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  • 1. ˜ ˆ ’ NGUYEN THUY THANH ` ˆ BAI TAP . ´ ´ ˆ TOAN CAO CAP Tˆp 3 a. e ınh ıch a y ´ e ˜ Ph´p t´ t´ phˆn. L´ thuyˆt chuˆ i. o Phu.o.ng tr` vi phˆn ınh a ` ´ ˆ ’ ´ ˆ ` ˆNHA XUAT BAN DAI HOC QUOC GIA HA NOI . . .
  • 2. Muc luc . . a ´ .10 T´ phˆn bˆt dinh ıch a 4 10.1 C´c phu.o.ng ph´p t´ t´ch phˆn . . . . . a a ınh ı a . . . . . . . 4 a ı a a .´ 10.1.1 Nguyˆn h`m v` t´ch phˆn bˆt dinh e a . . . . . . . 4 10.1.2 Phu.o.ng ph´p dˆi biˆn . . . . . . . a o e’ ´ . . . . . . . 12 10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn a ıch a u ` a . . . . . . . 21 10.2 C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp . . . . a o a ’ ıch o a a ´ a 30 10.2.1 T´ phˆn c´c h`m h˜.u ty . . . . . . . . . . . . ıch a a a u ’ 30 10.2.2 T´ phˆn mˆt sˆ h`m vˆ ty do.n gian . . . . . ıch a . ´ o o a o ’ ’ 37 10.2.3 T´ phˆn c´c h`m lu.o.ng gi´c . . . . . . . . . . ıch a a a . a 4811 T´ phˆn x´c dinh Riemann ıch a a . 57 ’ ıch 11.1 H`m kha t´ Riemann v` t´ch phˆn x´c dinh . . . a a ı a a . . . 58 -. 11.1.1 Dinh ngh˜ . . . . . . . . . . . . . . . . . . ıa . . 58 - ` e e e a . ’ ’ ı 11.1.2 Diˆu kiˆn dˆ h`m kha t´ch . . . . . . . . . . . . 59 a ınh a ´ 11.1.3 C´c t´ chˆt co ’ . ban cua t´ch phˆn x´c dinh ’ ı a a . . . 59 11.2 Phu.o.ng ph´p t´ t´ phˆn x´c d inh . . . . . . . a ınh ıch a a . . . 61 .ng dung cua t´ch phˆn x´c d inh . . . . . . . ´ 11.3 Mˆt sˆ u o o´ . ’ ı a a . . . 78 . ’ ’ 11.3.1 Diˆn t´ h` ph˘ng v` thˆ t´ch vˆt thˆ . . e ıch ınh a a e ı a. e’ . . 78 11.3.2 T´ dˆ d`i cung v` diˆn t´ m˘t tr`n xoay . . ınh o a . a e ıch a o . . 89 11.4 T´ phˆn suy rˆng . . . . . . . . . . . . . . . . . . . . ıch a o . 98 11.4.1 T´ phˆn suy rˆng cˆn vˆ han . . . . . . . . . 98 ıch a o . a o . . ıch a o . ’ a 11.4.2 T´ phˆn suy rˆng cua h`m khˆng bi ch˘n . . 107 o . a .
  • 3. 2 MUC LUC . . 12 T´ phˆn h`m nhiˆu biˆn ıch a a `e e´ 117 12.1 T´ phˆn 2-l´.p . . . . . . . . . . . . . . ıch a o . . . . . . . . 118 .`.ng ho.p miˆn ch˜. nhˆt . . . 12.1.1 Tru o ` e u a . . . . . . . . 118 . . .`.ng ho.p miˆn cong . . . . . . 12.1.2 Tru o ` e . . . . . . . . 118 . 12.1.3 Mˆt v`i u o a ´ .ng dung trong h` hoc ınh . . . . . . . . . 121 . . 12.2 T´ phˆn 3-l´ ıch a o.p . . . . . . . . . . . . . . . . . . . . . . 133 12.2.1 Tru.`.ng ho.p miˆn h`nh hˆp . . . o . ` ı e o . . . . . . . . . 133 .`.ng ho.p miˆn cong . . . . . . 12.2.2 Tru o ` e . . . . . . . . 134 . 12.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 136 12.2.4 Nhˆn x´t chung . . . . . . . . . . a e . . . . . . . . . 136 12.3 T´ phˆn d u.`.ng . . . . . . . . . . . . . ıch a o . . . . . . . . 144 12.3.1 C´c dinh ngh˜a co. ban . . . . . . a . ı ’ . . . . . . . . 144 12.3.2 T´ t´ phˆn du o ınh ıch a .`.ng . . . . . . . . . . . . . . 146 12.4 T´ phˆn m˘t . . . . . . . . . . . . . . ıch a a. . . . . . . . . 158 12.4.1 C´c dinh ngh˜a co. ban . . . . . . a . ı ’ . . . . . . . . 158 12.4.2 Phu.o.ng ph´p t´ t´ch phˆn m˘t a ınh ı a a . . . . . . . . . 160 12.4.3 Cˆng th´ o u.c Gauss-Ostrogradski . . . . . . . . . 162 12.4.4 Cˆng th´.c Stokes . . . . . . . . . o u . . . . . . . . 162 y ´ 13 L´ thuyˆt chuˆ i e ˜ o 177 13.1 Chuˆ i sˆ du.o.ng . . . . . . . . . . . . . . . . . . . . . . ˜ o o ´ 178 13.1.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . . a . ı ’ 178 ˜ o o ´ 13.1.2 Chuˆ i sˆ du .o.ng . . . . . . . . . . . . . . . . . . 179 ˜ o . . ´ . o e o . ´ 13.2 Chuˆ i hˆi tu tuyˆt d ˆi v` hˆi tu khˆng tuyˆt d ˆi . . . o . e o a o . 191 13.2.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . . a . ı ’ 191 ˜ o ´ a a a ´ 13.2.2 Chuˆ i dan dˆu v` dˆu hiˆu Leibnitz . . . . . . e . 192 ˜ u 13.3 Chuˆ i l˜y th` o u.a . . . . . . . . . . . . . . . . . . . . . . 199 13.3.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . . a . ı ’ 199 13.3.2 Diˆu kiˆn khai triˆn v` phu.o.ng ph´p khai triˆn - `e e . ’ e a a ’ e 201 ˜ 13.4 Chuˆ i Fourier . . . . . . . . . . . . . . . . . . . . . . . o 211 13.4.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . . a . ı ’ 211
  • 4. MUC LUC . . 3 13.4.2 Dˆu hiˆu du vˆ su. hˆi tu cua chuˆ i Fourier . . . 212 ´ a e . ’ ` . o . ’ e . ˜ o14 Phu.o.ng tr` vi phˆn ınh a 224 14.1 Phu.o.ng tr` vi phˆn cˆp 1 . . . . . . . . . . . . . . . 225 ınh a a ´ 14.1.1 Phu.o.ng tr` t´ch biˆn . . . . . . . . . . . . . . 226 ınh a ´ e 14.1.2 Phu .o.ng tr` d ang cˆp . . . . . . . . . . . . . 231 ınh ˘ ’ ´ a 14.1.3 Phu.o.ng tr` tuyˆn t´ . . . . . . . . . . . . . 237 ınh ´ e ınh 14.1.4 Phu.o.ng tr` Bernoulli . . . . . . . . . . . . . . 244 ınh 14.1.5 Phu .o.ng tr` vi phˆn to`n phˆn . . . . . . . . 247 ınh a a `a 14.1.6 Phu.o.ng tr` Lagrange v` phu.o.ng tr` Clairaut255 ınh a ınh 14.2 Phu .o.ng tr` vi phˆn cˆp cao . . . . . . . . . . . . . . 259 ınh a a ´ 14.2.1 C´c phu a .o.ng tr` cho ph´p ha thˆp cˆp . . . . 260 ınh e ´ ´ . a a 14.2.2 Phu.o.ng tr` vi phˆn tuyˆn t´ cˆp 2 v´.i hˆ ınh a ´ e ınh a´ o e . ´ ` sˆ h˘ng . . . . . . . . . . . . . . . . . . . . . . 264 o a 14.2.3 Phu.o.ng tr` vi phˆn tuyˆn t´nh thuˆn nhˆt ınh a ´ e ı ` a ´ a cˆp n (ptvptn cˆp n ) v´.i hˆ sˆ h˘ng . . . . . . 273 a´ ´ a o e o ` . ´ a .o.ng tr` vi phˆn tuyˆn t´ cˆp 1 v´.i hˆ sˆ h˘ng290 ´ ´ o e o ` 14.3 Hˆ phu e . ınh a e ınh a . ´ a15 Kh´i niˆm vˆ phu.o.ng tr` a e . ` e ınh vi phˆn dao h`m riˆng a . a e 304 15.1 Phu.o.ng tr` vi phˆn cˆp 1 tuyˆn t´ dˆi v´.i c´c dao ınh a a ´ ´ ´ e ınh o o a . h`m riˆng . . . . . . . . . . . . . . . . . . . . . . . . . a e 306 15.2 Giai phu.o.ng tr` d ao h`m riˆng cˆp 2 d o.n gian nhˆt ’ ınh . a e ´ a ’ ´ a 310 15.3 C´c phu.o.ng tr` vˆt l´ to´n co. ban . . . . . . . . . . a ınh a y a . ’ 313 15.3.1 Phu.o.ng tr` truyˆn s´ng . . . . . . . . . . . . ınh ` o e 314 15.3.2 Phu .o.ng tr` truyˆn nhiˆt . . . . . . . . . . . . ınh ` e e 317 . 15.3.3 Phu .o.ng tr` Laplace . . . . . . . . . . . . . . ınh 320 a e . ’ T`i liˆu tham khao . . . . . . . . . . . . . . . . . . . . . 327
  • 5. Chu.o.ng 10 ıch a ´T´ phˆn bˆt dinh a . 10.1 C´c phu.o.ng ph´p t´ a a ınh t´ phˆn . . . . . . ıch a 4 e a a ıch a a . ´ 10.1.1 Nguyˆn h`m v` t´ phˆn bˆt dinh . . . . . 4 10.1.2 Phu.o.ng ph´p dˆi biˆn . . . . . . . . . . . . 12 a o e’ ´ 10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn . . . . . 21 a ıch a u ` a 10.2 C´c l´.p h`m kha t´ trong l´.p c´c h`m a o a ’ ıch o a a . cˆp . . . . . . . . . . . . . . . . . . . . . . 30 so a ´ 10.2.1 T´ phˆn c´c h`m h˜.u ty . . . . . . . . . 30 ıch a a a u ’ 10.2.2 T´ phˆn mˆt sˆ h`m vˆ ty do.n gian . . . 37 ıch a . ´ o o a o ’ ’ 10.2.3 T´ phˆn c´c h`m lu.o.ng gi´c . . . . . . . 48 ıch a a a . a10.1 C´c phu.o.ng ph´p t´ a a ınh t´ phˆn ıch a10.1.1 a a ıch a ´ Nguyˆn h`m v` t´ phˆn bˆt dinh e a .Dinh ngh˜ 10.1.1. H`m F (x) du.o.c goi l` nguyˆn h`m cua h`m-. ıa a . . a e a ’ a ’ ´ ’ o a ’f (x) trˆn khoang n`o d´ nˆu F (x) liˆn tuc trˆn khoang d´ v` kha vi e a o e e . e
  • 6. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 5 ˜ ’ ’ ’tai mˆ i diˆm trong cua khoang v` F (x) = f(x). . o e aDinh l´ 10.1.1. (vˆ su. tˆn tai nguyˆn h`m) Moi h`m liˆn tuc trˆn-. y ` . ` . e o e a . a e . e . ` o e e a e ’doan [a, b] dˆu c´ nguyˆn h`m trˆn khoang (a, b).-. ´ a y ’ uDinh l´ 10.1.2. C´c nguyˆn h`m bˆt k` cua c`ng mˆt h`m l` chı y a e a o a a ’ . .i mˆt h˘ng sˆ cˆng.kh´c nhau bo a ’ . ` o a ´ . o o Kh´c v´.i dao h`m, nguyˆn h`m cua h`m so. cˆp khˆng phai bao a o . a e a ’ a ´ a o ’gi`. c˜ng l` h`m so. cˆp. Ch˘ng han, nguyˆn h`m cua c´c h`m e−x , 2 o u a a ´ a ’ a . e a ’ a a 1 cos x sin xcos(x2), sin(x2), , , ,... l` nh˜.ng h`m khˆng so. cˆp. a u a o ´ a lnx x xD.nh ngh˜ 10.1.2. Tˆp ho.p moi nguyˆn h`m cua h`m f (x) trˆn-i ıa a . . . e a ’ a e ’ .o.c goi l` t´ phˆn bˆt dinh cua h`m f (x) trˆn khoangkhoang (a, b) du . . a ıch a a . ´ ’ a e ’(a, b) v` du.o.c k´ hiˆu l` a . y e a . f(x)dx. ´ a o . a e a ’ a e ’ Nˆu F (x) l` mˆt trong c´c nguyˆn h`m cua h`m f(x) trˆn khoang e(a, b) th` theo dinh l´ 10.1.2 ı . y f(x)dx = F (x) + C, C∈Rtrong d´ C l` h˘ng sˆ t`y y v` d˘ng th´.c cˆn hiˆu l` d˘ng th´.c gi˜.a o a ` a ´ o u ´ a a ’ u ` a ’ e a a ’ u uhai tˆp ho.p. a . . C´c t´ chˆt co. ban cua t´ phˆn bˆt dinh: a ınh a ´ ’ ’ ıch a a . ´ 1) d f (x)dx = f (x)dx. 2) f (x)dx = f (x). 3) df(x) = f (x)dx = f(x) + C. T`. dinh ngh˜ t´ phˆn bˆt dinh r´t ra bang c´c t´ch phˆn co. u . ıa ıch a ´ a . u ’ a ı aban (thu.`.ng du.o.c goi l` t´ phˆn bang) sau dˆy: ’ o . . a ıch a ’ a
  • 7. 6 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . I. 0.dx = C. II. 1dx = x + C. xα+1 III. xαdx = + C, α = −1 α+1 dx IV. = ln|x| + C, x = 0. x ax V. axdx = + C (0 < a = 1); ex dx = ex + C. lna VI. sin xdx = − cos x + C. VII. cos xdx = sin x + C. dx π VIII. 2x = tgx + C, x = + nπ, n ∈ Z. cos 2 dx IX. = −cotgx + C, x = nπ, n ∈ Z. sin2 x  dx arc sin x + C, X. √ = −1 < x < 1. 1 − x2 −arc cos x + C  dx arctgx + C, XI. = 1 + x2 −arccotgx + C. dx √ XII. √ = ln|x + x2 ± 1| + C x2 ± 1 (trong tru.`.ng ho.p dˆu tr`. th` x < −1 ho˘c x > 1). o . ´ a u ı a . dx 1 1+x XIII. 2 = ln + C, |x| = 1. 1−x 2 1−x ´ ınh ıch a a .´ C´c quy t˘c t´ t´ phˆn bˆt dinh: a a
  • 8. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 7 1) kf(x)dx = k f(x)dx, k = 0. 2) [f(x) ± g(x)]dx = f (x)dx ± g(x)dx. ´ 3) Nˆu e ’ f(x)dx = F (x) + C v` u = ϕ(x) kha vi liˆn tuc th` a e . ı f (u)du = F (u) + C. CAC V´ DU ´ I .V´ du 1. Ch´.ng minh r˘ng h`m y = signx c´ nguyˆn h`m trˆn ı . u ` a a o e a e ’ ´khoang bˆt k` khˆng ch´ a y o u .a diˆm x = 0 v` khˆng c´ nguyˆn h`m trˆn ’ e a o o e a emoi khoang ch´.a diˆm x = 0. . ’ u ’ e Giai. 1) Trˆn khoang bˆt k` khˆng ch´.a diˆm x = 0 h`m y = signx ’ e ’ ´ a y o u ’ e a ` ´ ’l` h˘ng sˆ. Ch˘ng han v´ a a o a .i moi khoang (a, b), 0 < a < b ta c´ signx = 1 ’ . o . o o . e a ’ o ev` do d´ moi nguyˆn h`m cua n´ trˆn (a, b) c´ dang a o . F (x) = x + C, C ∈ R. e ’ a e ’ 2) Ta x´t khoang (a, b) m` a < 0 < b. Trˆn khoang (a, 0) moi . e a ’ o . o e ’nguyˆn h`m cua signx c´ dang F (x) = −x + C1 c`n trˆn khoang (0, b)nguyˆn h`m c´ dang F (x) = x + C2. V´.i moi c´ch chon h˘ng sˆ C1 e a o . o . a . ` a ´ ov` C2 ta thu du.o.c h`m [trˆn (a, b)] khˆng c´ dao h`m tai diˆm x = 0. a . a e o o . a ’ . eNˆu ta chon C = C1 = C2 th` thu du.o.c h`m liˆn tuc y = |x| + C e´ . ı . a e .nhu.ng khˆng kha vi tai diˆm x = 0. T`. d´, theo dinh ngh˜a 1 h`m o ’ . e’ u o . ı asignx khˆng c´ nguyˆn h`m trˆn (a, b), a < 0 < b. o o e a eV´ du 2. T` nguyˆn h`m cua h`m f (x) = e|x| trˆn to`n truc sˆ. ı . ım e a ’ a e a . o ´ ’ Giai. V´ o.i x |x| x ` 0 ta c´ e = e v` do d´ trong miˆn x > 0 mˆt o a o e o .trong c´c nguyˆn h`m l` ex . Khi x < 0 ta c´ e|x| = e−x v` do vˆy a e a a o a a . e o a e a a −xtrong miˆn x < 0 mˆt trong c´c nguyˆn h`m l` −e + C v´ ` ` .i h˘ng o a . ´ ´sˆ C bˆt k`. o a y Theo dinh ngh˜ nguyˆn h`m cua h`m e|x| phai liˆn tuc nˆn n´ . ıa, e a ’ a ’ e . e o
  • 9. 8 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . ’ ’ a ` phai thoa m˜n diˆu kiˆn e e . lim ex = lim (−e−x + C) x→0+0 x→0−0 t´.c l` 1 = −1 + C ⇒ C = 2. u a Nhu. vˆy a .  ex  ´ nˆu x > 0, e   F (x) = 1 ´ nˆu x = 0, e    −x −e + 2 nˆu x < 0 ´ e l` h`m liˆn tuc trˆn to`n truc sˆ. Ta ch´.ng minh r˘ng F (x) l` nguyˆn a a e . e a . o ´ u ` a a e ’ |x| ´ h`m cua h`m e trˆn to`n truc sˆ. Thˆt vˆy, v´ a a e a .i x > 0 ta c´ . o a a . . o o F (x) = ex = e|x|, v´.i x < 0 th` F (x) = e−x = e|x|. Ta c`n cˆn phai o ı o ` a ’ ch´.ng minh r˘ng F (0) = e0 = 1. Ta c´ u ` a o F (x) − F (0) ex − 1 F+ (0) = lim = lim = 1, x→0+0 x x→0+0 x F (x) − F (0) −e−x + 2 − 1 F− (0) = lim = lim = 1. x→0−0 x x→0−0 x Nhu. vˆy F+ (0) = F− (0) = F (0) = 1 = e|x|. T`. d´ c´ thˆ viˆt: a . u o o e e ’ ´  ex + C, x<0 e|x|dx = F (x) + C = −e−x + 2 + C, x < 0. V´ du 3. T` nguyˆn h`m c´ dˆ thi qua diˆm (−2, 2) dˆi v´.i h`m ı . ım e a o ` . o ’ e ´ o o a 1 f (x) = , x ∈ (−∞, 0). x 1 ’ Giai. V` (ln|x|) = nˆn ln|x| l` mˆt trong c´c nguyˆn h`m cua ı e a o . a e a ’ x 1 a . e a ’ h`m f(x) = . Do vˆy, nguyˆn h`m cua f l` h`m F (x) = ln|x| + C, a a a x C ∈ R. H˘ng sˆ C du.o.c x´c dinh t`. diˆu kiˆn F (−2) = 2, t´.c l` ` a ´ o . a . u ` e e . u a ln2 + C = 2 ⇒ C = 2 − ln2. Nhu a. vˆy . x F (x) = ln|x| + 2 − ln2 = ln + 2. 2
  • 10. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 9V´ du 4. T´ c´c t´ phˆn sau dˆy: ı . ınh a ıch a a 2x+1 − 5x−1 2x + 3 1) dx, 2) dx. 10x 3x + 2 ’ Giai. 1) Ta c´ o 2x 5x 1 x 1 1 x I= 2 x − x dx = 2 − dx 10 5 · 10 5 5 2 1 x 1 1 x =2 dx − dx 5 5 2 1 x 1 x 1 2 =2 5 − +C 1 5 1 ln ln 5 2 2 1 =− x + + C. 5 ln5 5 · 2x ln2 2) 3 2 5 2 x+ x+ + I= 2 dx = 2 3 6 dx 2 3 2 3 x+ x+ 3 3 2 5 2 = x + ln x + + C. 3 9 3V´ du 5. T´ c´c t´ phˆn sau dˆy: ı . ınh a ıch a a 1 + cos2 x √1) tg2 xdx, 2) dx, 3) 1 − sin 2xdx. 1 + cos 2x ’ Giai. 1) 2 sin2 x 1 − cos2 x tg xdx = dx = dx cos2 x cos2 x dx = − dx = tgx − x + C. cos2 x
  • 11. 10 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 2) 1 + cos2 x 1 + cos2 x 1 dx dx = 2x dx = + dx 1 + cos 2x 2 cos 2 cos2 x 1 = (tgx + x) + C. 2 3) √ 1 − sin 2xdx = sin2 x − 2 sin x cos x + cos2 xdx = (sin x − cos x)2dx = | sin x − cos x|dx = (sin x + cos x)sign(cos x − sin x) + C. ` ˆ BAI TAP . B˘ng c´c ph´p biˆn dˆi dˆng nhˆt, h˜y du.a c´c t´ch phˆn d˜ cho ` a a e ´ e o `’ o ´ a a a ı a a vˆ t´ phˆn bang v` t´ c´c t´ch phˆn d´1 ` ıch a e ’ a ınh a ı a o dx 1 x−1 1 1. . (DS. ln − arctgx) x4 − 1 4 x+1 2 1 + 2x2 1 2. dx. (DS. arctgx − ) x2 (1 + x2 ) x √ √ x2 + 1 + 1 − x2 √ 3. √ dx. (DS. arc sin x + ln|x + 1 + x2|) 1 − x4 √ √ x2 + 1 − 1 − x2 √ √ 4. √ dx. (DS. ln|x + x2 − 1| − ln|x + x2 + 1|) x4 − 1 √ x4 + x−4 + 2 1 5. 3 dx. (DS. ln|x| − 4 ) x 4x 23x − 1 e2x 6. dx. (DS. + ex + 1) ex − 1 2 Dˆ cho gon, trong c´c “D´p sˆ” cua chu.o.ng n`y ch´ng tˆi bo qua khˆng viˆt 1’ e . a ´ a o ’ a u o ’ o ´ e `ng sˆ cˆng C. c´c h˘ a a ´ . o o
  • 12. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 11 3x 22x − 1 2 22 x7. √ dx. (DS. + 2− 2 ) 2x ln2 3 dx 1 lnx8. . (DS. √ arctg √ ) x(2 + ln2 x) 2 2 √ 3 ln2 x 3 5/39. dx. (DS. ln x) x 5 ex + e2x10. dx. (DS. −ex − 2ln|ex − 1|) 1 − ex ex dx11. . (DS. ln(1 + ex)) 1 + ex x 1 sin x12. sin2 dx. (DS. x− ) 2 2 213. cotg2 xdx. (DS. −x − cotgx) √ π14. 1 + sin 2xdx, x ∈ 0, . (DS. − cos x + sin x) 215. ecos x sin xdx. (DS. −ecos x )16. ex cos ex dx. (DS. sin ex) 1 x17. dx. (DS. tg ) 1 + cos x 2 dx 1 x π18. . (DS. √ ln tg + ) sin x + cos x 2 2 8 1 + cos x 219. dx. (DS. − ) (x + sin x)3 2(x + sin x)2 sin 2x 120. dx. (DS. − 1 − 4 sin2 x) 1 − 4 sin x 2 2 sin x √21. dx. (DS. −ln| cos x + 1 + cos2 x|) 2 2 − sin x
  • 13. 12 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . sin x cos x 1 sin2 x 22. dx. (DS. arc sin √ ) 3 − sin4 x 2 3 arccotg3x 1 23. 2 dx. (DS. − arccotg2 3x) 1 + 9x 6 √ x + arctg2x 1 1 24. dx. (DS. ln(1 + 4x2) + arctg3/22x) 1 + 4x2 8 3 arc sin x − arc cos x 1 25. √ dx. (DS. (arc sin2 x + arc cos2 x)) 1 − x2 2 x + arc sin3 2x 1√ 1 26. √ dx. (DS. − 1 − 4x2 + arc sin4 2x) 1 − 4x2 4 8 x + arc cos3/2 x √ 2 27. √ dx. (DS. − 1 − x2 − arc cos5/2 x) 1 − x2 5 |x|3 28. x|x|dx. (DS. ) 3 29. (2x − 3)|x − 2|dx.  − 2 x3 + 7 x2 − 6x + C, x < 2  (DS. F (x) = 3 2 ) 2 3 7 2  x − x + 6x + C, x 2 3 2  1 − x2, |x| 1, 30. f(x)dx, f(x) = 1 − |x|, |x| > 1.  3 x − x + C  ´ nˆu |x| e 1 (DS. F (x) = 3 ) x − x|x| + 1 signx + C  ´ nˆu|x| > 1 e 2 6 10.1.2 Phu.o.ng ph´p dˆi biˆn a o’ ´ e Dinh l´. Gia su.: -. y ’ ’
  • 14. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 13 1) H`m x = ϕ(t) x´c dinh v` kha vi trˆn khoang T v´.i tˆp ho.p gi´ a a . a ’ e ’ o a .. a ’tri l` khoang X. . a a . a o e a e ’ 2) H`m y = f (x) x´c dinh v` c´ nguyˆn h`m F (x) trˆn khoang X. a o a a e a ’ Khi d´ h`m F (ϕ(t)) l` nguyˆn h`m cua h`m f(ϕ(t))ϕ (t) trˆn a e ’khoang T . T`. dinh l´ 10.1.1 suy r˘ng u . y ` a f (ϕ(t))ϕ (t)dt = F (ϕ(t)) + C. (10.1) V` ı F (ϕ(t)) + C = (F (x) + C) x=ϕ(t) = f (x)dx x=ϕ(t)cho nˆn d˘ng th´.c (10.1) c´ thˆ viˆt du.´.i dang e a ’ u ’ ´ o e e o . f(x)dx x=ϕ(t) = f (ϕ(t))ϕ (t)dt. (10.2) D˘ng th´.c (10.2) du.o.c goi l` cˆng th´.c dˆi biˆn trong t´ phˆn ’ a u . . a o u o e ’ ´ ıch a ´bˆt dinh. a . Nˆu h`m x = ϕ(t) c´ h`m ngu.o.c t = ϕ−1 (x) th` t`. (10.2) thu ´ e a o a . ı u .o.cdu . f(x)dx = f (ϕ(t))ϕ (t)dt t=ϕ−1 (x) . (10.3) e o a ı . ` e o e Ta nˆu mˆt v`i v´ du vˆ ph´p dˆi biˆn. . e ’ ´ √ i) Nˆu biˆu th´.c du.´.i dˆu t´ phˆn c´ ch´.a c˘n a2 − x2, a > 0 ´ e e’ u ´ o a ıch a o u a π πth` su. dung ph´p dˆi biˆn x = a sin t, t ∈ − , ı ’ . e o e’ ´ . 2 2 √ ii) Nˆu biˆu th´.c du.´.i dˆu t´ phˆn c´ ch´.a c˘n x2 − a2, a > 0 ´ e e’ u ´ o a ıch a o u a a π e o e ’ ´th` d`ng ph´p dˆi biˆn x = ı u , 0 < t < ho˘c x = acht. a . cos t 2 √ ´ .´.i dˆu t´ phˆn ch´.a c˘n th´.c a2 + x2, a > 0 iii) Nˆu h`m du o a ıch a e a ´ u a u π π ’ .th` c´ thˆ d˘t x = atgt, t ∈ − , ı o e a ho˘c x = asht. a . 2 2 ´ .´.i dˆu t´ phˆn l` f (x) = R(ex , e2x, . . . .enx ) th` iv) Nˆu h`m du o a ıch a a e a ´ ıc´ thˆ d˘t t = ex (o. dˆy R l` h`m h˜.u ty). o e a ’ . ’ a a a u ’
  • 15. 14 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . CAC V´ DU ´ I . dx V´ du 1. T´ ı . ınh . cos x ’ Giai. Ta c´ o dx cos xdx = (d˘t t = sin x, dt = cos xdx) a . cos x 1 − sin2 x dt 1 1+t x π = = ln + C = ln tg + + C. 1 − t2 2 1−t 2 4 x3 dx V´ du 2. T´ I = ı . ınh . x8 − 2 ’ Giai. ta c´ o √ 1 2 x4 d(x4 ) d √ 4 4 2 I= = x8 − 2 x4 2 −2 1 − √ 2 x4 D˘t t = √ ta thu du.o.c a . . 2 √ √ 2 2 + x4 I=− ln √ + C. 8 2 − x4 x2 dx V´ du 3. T´ I = ı . ınh · (x2 + a2 )3 adt ’ Giai. D˘t x(t) = atgt ⇒ dx = a . . Do d´ o cos2 t a3tg2t · cos3 tdt sin2 t dt I= = dt = − cos tdt a3 cos2 t cos t cos t t π = ln tg + − sin t + C. 2 4 x V` t = arctg nˆn ı e a 1 x π x I = ln tg arctg + − sin arctg +C 2 a 4 a x √ = −√ + ln|x + x2 + a2| + C. x2 + a2
  • 16. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 15 . . ı e ˜ a e ´ `Thˆt vˆy, v` sin α = cos α · tgα nˆn dˆ d`ng thˆy r˘ng a a a a x x sin arctg =√ · a x 2 + a2 ´ Tiˆp theo ta c´ e o 1 x π x π x sin arctg + 1 − cos arctg + 1 + sin arctg 2 a 4 = a 2 = a 1 x π x π x cos arctg + sin arctg + − cos arctg 2 a 4 a 2 a √ x + a2 + x2 = av` t`. d´ suy ra diˆu phai ch´.ng minh. a u o ` e ’ u √V´ du 4. T´ I = ı . ınh a2 + x2 dx. ’ Giai. D˘t x = asht. Khi d´ a . o I= a2 (1 + sh2 t)achtdt = a2 ch2 tdt ch2t + 1 a2 1 = a2 dt = sh2t + t + C 2 2 2 a2 = (sht · cht + t) + C. 2 √ 2 x2 t x+ a2 + x2V` cht = ı 1 + sh t = 1 + 2 . e = sht + cht = nˆn e √ a a x + a2 + x2t = ln v` do d´ a o a √ x√ 2 a2 √ a2 + x2 dx = a + x2 + ln|x + a2 + x2| + C. 2 2V´ du 5. T´ ı . ınh x2 + 1 3x + 4 1) I1 = √ dx, 2) I2 = √ dx. x6 − 7x4 + x2 −x2 + 6x − 8
  • 17. 16 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . ’ Giai. 1) Ta c´ o 1 1 1+ d x− dt I1 = x2 dx = x = √ 1 1 2 t2 − 5 x2 − 7 + x− −5 x2 x √ 1 1 = ln|t + t2 − 5| + C = ln x − + x2 − 7 + 2 + C. x x 2) Ta viˆt biˆu th´.c du.´.i dˆu t´ phˆn du.´.i dang ´ e e ’ u ´ o a ıch a o . 3 −2x + 6 1 f (x) = − · √ + 13 · √ 2 −x2 + 6x − 8 −x2 + 6x − 8 v` thu du.o.c a . I2 = f(x)dx 3 1 d(x − 3) =− (−x2 + 6x − 8)− 2 d(−x2 + 6x − 8) + 13 2 1 − (x − 3)2 √ = −3 −x2 + 6x − 8 + 13 arc sin(x − 3) + C. V´ du 6. T´ ı . ınh dx sin x cos3 x 1) , 2) I2 = dx. sin x 1 + cos2 x ’ Giai 1) C´ch I. Ta c´ a o dx sin x d(cos x) 1 1 − cos x = dx = = ln + C. sin x sin2 x cos2 x − 1 2 1 + cos x C´ch II. a x x dx d d = 2 2 sin x x x = x x sin cos tg · cos2 2 2 2 2 x d tg x 2 = x = ln tg 2 + C. tg 2
  • 18. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 17 2) Ta c´ o sin x cos x[(cos2 x + 1) − 1] I2 = dx. 1 + cos2 x Ta d˘t t = 1 + cos2 x. T`. d´ dt = −2 cos x sin xdx. Do d´ a . u o o 1 t−1 t I2 = − dt = − + ln|t| + C, 2 t 2trong d´ t = 1 + cos2 x. oV´ du 7. T´ ı . ınh exdx ex + 1 1) I1 = √ , 2) I2 = dx. e2x + 5 ex − 1 ’ Giai 1) D˘t ex = t. Ta c´ ex dx = dt v` a . o a dt √ √ I1 = √ = ln|t + t2 + 5| + C = ln |ex + e2x + 5| + C. t2 + 5 dt 2) Tu.o.ng tu., d˘t ex = t, exdx = dt, dx = . a . v` thu du.o.c a . t t + 1 dt 2dt dt I2 = = − = 2ln|t − 1| − ln|t| + C t−1 t t−1 t = 2ln|ex − 1| − lnex + c = ln(ex − 1)2 − x + C. ` ˆ BAI TAP . T´ c´c t´ phˆn: ınh a ıch a e2x 41. √ 4 dx. (DS. (3ex − 4) 4 (ex + 1)3 ) ex+1 21 ’ ˜ Chı dˆ n. D˘t ex + 1 = t4. a a .
  • 19. 18 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . √ dx 1 + ex − 1 2. √ . (DS. ln √ ) ex + 1 1 + ex + 1 e2x 3. dx. (DS. ex + ln|ex − 1|) ex − 1 √ 1 + lnx 2 4. dx. (DS. (1 + lnx)3) x 3 √ 1 + lnx 5. dx. xlnx √ √ (DS. 2 1 + lnx − ln|lnx| + 2ln| 1 + lnx − 1|) dx x x 6. . (DS. −x − 2e− 2 + 2ln(1 + e 2 )) ex/2+e x √ arctg x dx √ 7. √ . (DS. (arctg x)2) x 1+x √ 2 8. e3x + e2xdx. (DS. (ex + 1)3/2 ) 3 2 +2x−1 1 2x2+2x−1 9. e2x (2x + 1)dx. (DS. e ) 2 dx √ 10. √ . (DS. 2arctg ex − 1) ex − 1 e2xdx 1 √ 11. √ . (DS. ln(e2x + e4x + 1)) e4x + 1 2 2x dx arc sin 2x 12. √ . (DS. ) 1 − 4x ln2 dx √ √ 13. √ . (DS. 2[ x + 1 − ln(1 + x + 1)]) 1+ x+1 ’ ˜ Chı dˆ n. D˘t x + 1 = t2. a a . x+1 √ √ x−2 14. √ dx. (DS. 2 x − 2 + 2arctg ) x x−2 2 dx 2 √ √ 15. √ . (DS. ax + b − mln| ax + b + m| ) ax + b + m a
  • 20. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 19 dx √ √16. √ √ . (DS. 3 3 x + 3ln| 3 x − 1|) 3 x( x − 1) 3 dx17. . (DS. tg(arc sin x)) (1 − x2)3/2 π π ’ a˜ Chı dˆ n. D˘t x = sin t, t ∈ a . − , ) 2 2 dx 1 x18. . (DS.sin arctg ) (x2 + a2)3/2 a2 a π π ’ a˜ Chı dˆ n. D˘t x = atgt, t ∈ − , a . . 2 2 dx 1 119. . (DS. − , t = arc sin ) (x2 − 1)3/2 cos t x 1 π π ’ a˜ Chı dˆ n. D˘t x = a . , − < t < 0, 0 < t < . sin t 2 2 √ √ a2 x x a2 − x220. a2 − x2 dx. (DS. arc sin + ) 2 a 2 ’ ˜ Chı dˆ n. D˘t x = a sin t. a a . √ x√ 2 a2 √21. a2 + x2dx. (DS. a + x2 + ln|x + a2 + x2|) 2 2 ’ ˜ Chı dˆ n. D˘t x = asht. a a . x2 1 √ 2 √22. √ dx. (DS. x a + x2 − a2ln(x + a2 + x2) ) a2 + x2 2 √ dx x2 + a223. √ . (DS. − ) x2 x2 + a2 a2x 1 ’ ˜ Chı dˆ n. D˘t x = a a . ho˘c x = atgt, ho˘c x = asht. a . a . t x2dx a2 x x√ 224. √ . (DS. arc sin − a − x2 ) a2 − x2 2 a a ’ ˜ Chı dˆ n. D˘t x = a sin t. a a . dx 1 a25. √ . (DS. − arc sin ) x x2 − a2 a x
  • 21. 20 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 1 a ’ a˜ Chı dˆ n. D˘t x = , ho˘c x = a. a . ho˘c x = acht. a . t cos t √ √ 1 − x2 1 − x2 26. dx. (DS. − − arc sin x) x2 x dx x 27. . (DS. √ ) (a2 + x2)3 a2 x2 + a2 √ dx x2 − 9 28. √ . (DS. ) x 2 x2 − 9 9x dx x 29. . (DS. − √ ) (x2 − a2)3 a2 x2 − a2 √ 30. x2 a2 − x2dx. x a2 √ a4 x (DS. − (a2 − x2)3/2 + x x2 − a2 + arc sin ) 4 8 8 a a+x √ x 31. dx. (DS. − a2 − x2 + arc sin ) a−x a ’ a˜ n. D˘t x = a cos 2t. Chı dˆ a . x−a 32. dx. x+a √ √ √ (DS. ´ x2 − a2 − 2aln( x − a + x + a) nˆu x > a, e √ √ √ ´ − x2 − a2 + 2aln( −x + a + −x − a) nˆu x < −a) e a ’ a˜ Chı dˆ n. D˘t x = a . . cos 2t √ x − 1 dx 1 x2 − 1 33. . (DS. arc cos − ) x + 1 x2 x x 1 ’ a˜ Chı dˆ n. D˘t x = . a . t dx √ 34. √ . (DS. 2arc sin x) x − x2
  • 22. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 21 Chı dˆ n. D˘t x = sin2 t. ’ a˜ a. √ √ x2 + 1 √ 1 + x2 + 135. dx. (DS. x2 + 1 − ln ) x x x3dx x2 √ 4√36. √ . (DS. − 2 − x2 − 2 − x2) 2 − x2 3 3 (9 − x2)2 (9 − x2 )537. dx. (DS. − ) x6 45x5 x2dx x√ 2 a2 √38. √ . (DS. x − a2 + ln|x + x2 − a2|) x2 − a2 2 2 (x + 1)dx xex39. . (DS. ln ) x(1 + xex) 1 + xex Chı dˆ n. Nhˆn tu. sˆ v` mˆ u sˆ v´.i ex rˆi d˘t xex = t. ’ ˜ a a ’ o a ˜ o o ´ a ´ ` a o . dx 1 x ax40. . (DS. 3 arctg + 2 ) (x2 + a2)2 2a a x + a2 ’ ˜ Chı dˆ n. D˘t x = atgt. a a.10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn a ıch a u ` aPhu.o.ng ph´p t´ phˆn t`.ng phˆn du.a trˆn dinh l´ sau dˆy. a ıch a u ` a . e . y aD.nh l´. Gia su. trˆn khoang D c´c h`m u(x) v` v(x) kha vi v` h`m-i y ’ ’ e ’ a a a ’ a av(x)u (x) c´ nguyˆn h`m. Khi d´ h`m u(x)v (x) c´ nguyˆn h`m trˆn o e a o a o e a eD v` a u(x)v (x)dx = u(x)v(x) − v(x)u (x)dx. (10.4) Cˆng th´.c (10.4) du.o.c goi l` cˆng th´.c t´nh t´ phˆn t`.ng phˆn. o u . . a o u ı ıch a u ` aV` u (x)dx = du v` v (x)dx = dv nˆn (10.4) c´ thˆ viˆt du.´.i dang ı a e o e e’ ´ o . udv = uv − vdu. (10.4*) Thu.c tˆ cho thˆy r˘ng phˆn l´.n c´c t´ phˆn t´nh du.o.c b˘ng . e ´ ´ ` a a ` o a a ıch a ı . ` aph´p t´ phˆn t`.ng phˆn c´ thˆ phˆn th`nh ba nh´m sau dˆy. e ıch a u ` o e a a ’ a o a
  • 23. 22 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . Nh´m I gˆm nh˜.ng t´ch phˆn m` h`m du.´.i dˆu t´ phˆn c´ ch´.a o ` o u ı a a a ´ o a ıch a o u .a sˆ l` mˆt trong c´c h`m sau dˆy: lnx, arc sin x, arc cos x, arctgx, u ´ th` o a o a a a . (arctgx)2, (arc cos x)2, lnϕ(x), arc sin ϕ(x),... Dˆ t´ c´c t´ phˆn n`y ta ´p dung cˆng th´.c (10.4*) b˘ng c´ch ’ e ınh a ıch a a a . o u ` a a . ` o . a a a ’ d˘t u(x) b˘ng mˆt trong c´c h`m d˜ chı ra c`n dv l` phˆn c`n lai cua a a o a ` o . ’ a biˆu th´.c du.´.i dˆu t´ phˆn. e’ u ´ o a ıch a Nh´m II gˆm nh˜.ng t´ phˆn m` biˆu th´.c du.´.i dˆu t´ phˆn o ` o u ıch a a e ’ u ´ o a ıch a c´ dang P (x)e , P (x) cos bx, P (x) sin bx trong d´ P (x) l` da th´.c, a, o . ax o a u a ` b l` h˘ng sˆ. a ´ o ’ Dˆ t´ c´c t´ phˆn n`y ta ´p dung (10.4*) b˘ng c´ch d˘t u(x) = e ınh a ıch a a a . ` a a a . P (x), dv l` phˆn c`n lai cua biˆu th´.c du.´.i dˆu t´ phˆn. Sau mˆ i a ` o . ’ a e’ u ´ o a ıch a ˜ o lˆn t´ phˆn t`.ng phˆn bˆc cua da th´.c s˜ giam mˆt do.n vi. ` ıch a u a ` a a ’ . u e ’ o . . Nh´m III gˆm nh˜ o `o u .ng t´ch phˆn m` h`m du.´.i dˆu t´ch phˆn c´ ı a a a o a ı ´ a o ax ax ` ıch a dang: e sin bx, e cos bx, sin(lnx), cos(lnx),... Sau hai lˆn t´ phˆn . a t`.ng phˆn ta lai thu du.o.c t´ch phˆn ban dˆu v´.i hˆ sˆ n`o d´. D´ l` u `a . . ı a ` . ´ a o e o a o o a phu .o.ng tr` tuyˆn t´ v´.i ˆn l` t´ phˆn cˆn t´ ınh ´ ’ e ınh o a a ıch a ` ınh. a Du .o.ng nhiˆn l` ba nh´m v`.a nˆu khˆng v´t hˆt moi t´ch phˆn e a o u e o e e ´ . ı a t´ du.o.c b˘ng t´ phˆn t`.ng phˆn (xem v´ du 6). ınh . ` a ıch a u ` a ı . Nhˆn x´t. Nh` a a e o. c´c phu.o.ng ph´p dˆi biˆn v` t´ phˆn t`.ng phˆn ’ ´ a o e a ıch a u ` a . ta ch´.ng minh du.o.c c´c cˆng th´.c thu.`.ng hay su. dung sau dˆy: u . a o u o ’ . a dx 1 x 1) = arctg + C, a = 0. x2 +a 2 a a dx 1 a+x 2) = ln + C, a = 0. a2 −x 2 2a a − x dx x 3) √ = arc sin + C, a = 0. a2 − x2 a dx √ 4) √ = ln|x + x2 ± a2| + C. x2 ± a2
  • 24. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 23 CAC V´ DU ´ I . √ √V´ du 1. T´ t´ phˆn I = ı . ınh ıch a xarctg xdx. ’ Giai. T´ phˆn d˜ cho thuˆc nh´m I. Ta d˘t ıch a a o . o a . √ u(x) = arctg x, √ dv = xdx. 1 dx 2 3Khi d´ du = o · √ , v = x 2 . Do d´ o 1+x 2 x 3 2 3 √ 1 x I = x 2 arctg x − dx 3 3 1+x 2 3 √ 1 1 = x 2 arctg x − 1− dx 3 3 1+x 2 3 √ 1 = x 2 arctg x − (x − ln|1 + x|) + C. 3 3V´ du 2. T´ I = arc cos2 xdx. ı . ınh Giai. Gia su. u = arc cos2 x, dv = dx. Khi d´ ’ ’ ’ o 2arc cos x du = − √ dx, v = x. 1 − x2Theo (10.4*) ta c´ o xarc cos x I = xarc cos2 x + 2 √ dx. 1 − x2 Dˆ t´ t´ phˆn o. vˆ phai d˘ng th´.c thu du.o.c ta d˘t u = ’ e ınh ıch a ’ e ´ ’ ’ a u . a . xdxarc cos x, dv = √ . Khi d´ o 1 − x2 dx √ √ du = − √ , v = − d( 1 − x2) = − 1 − x2 + C1 1 − x2 √ ’ ` a a ´v` ta chı cˆn lˆy v = − 1 − x2: a xarc cos x √ √ dx = − 1 − x2arc cos x − dx 21 − x2 √ = − 1 − x2arc cos x − x + C2 .
  • 25. 24 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . Cuˆi c`ng ta thu du.o.c ´ o u . √ I = xarc cos2 x − 2 1 − x2arc cos x − 2x + C. V´ du 3. T´ I = ı . ınh x2 sin 3xdx. ’ Giai. T´ phˆn d˜ cho thuˆc nh´m II. Ta d˘t ıch a a o . o a . u(x) = x2, dv = sin 3xdx. 1 Khi d´ du = 2xdx, v = − cos 3x v` o a 3 1 2 1 2 I = − x2 cos 3x + x cos 3xdx = − x2 cos 3x + I1. 3 3 3 3 ` ınh Ta cˆn t´ I1. D˘t u = x, dv = cos 3xdx. Khi d´ du = 1dx, a a . o 1 v = sin 3x. T`. d´ u o 3 1 2 1 1 I = − x2 cos 3x + x sin 3x − sin 3xdx 3 3 3 3 1 2 2 = − x2 cos 3x + x sin 3x + cos 3x + C. 3 9 27 Nhˆn x´t. Nˆu d˘t u = sin 3x, dv = x2dx th` lˆn t´ phˆn t`.ng a e . ´ . e a ı ` ıch a u a phˆn th´. nhˆt khˆng du.a dˆn t´ phˆn do.n gian ho.n. ` a u ´ a o ´ e ıch a ’ V´ du 4. T´ I = ı . ınh eax cos bx; a, b = 0. Giai. Dˆy l` t´ phˆn thuˆc nh´m III. Ta d˘t u = eax, dv = ’ a a ıch a o . o a . 1 cos bxdx. Khi d´ du = aeaxdx, v = sin bx v` o a b 1 a 1 a I = eax sin bx − eax sin bxdx = eax sin bx − I1 . b b b b ’ Dˆ t´ I1 ta d˘t u = eax, dv = sin bxdx. Khi d´ du = aeaxdx, e ınh a. o 1 v = − cos bx v` a b 1 a I1 = − eax cos bx + eax cos bxdx. b b
  • 26. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 25Thˆ I1 v`o biˆu th´.c dˆi v´.i I ta thu du.o.c ´ e a e’ u o o´ . 1 a a2 eax cos bxdx = eax sin bx + 2 cos bx − 2 eax cos bxdx. b b b Nhu. vˆy sau hai lˆn t´ phˆn t`.ng phˆn ta thu du.o.c phu.o.ng a . ` ıch a u a ` a .tr` tuyˆn t´ v´.i ˆn l` I. Giai phu.o.ng tr` thu du.o.c ta c´ ınh ´ e ınh o a a ’ ’ ınh . o a cos bx + b sin bx eax cos bxdx = eax + C. a2 + b2V´ du 5. T´ I = ı . ınh sin(ln x)dx. 1 ’ Giai. D˘t u = sin(lnx), dv = dx. Khi d´ du = cos(lnx)dx, a. o xv = x. Ta thu du.o.c . I = x sin(lnx) − cos(lnx)dx = x sin(lnx) − I1. ’ Dˆ t´ e ınh I1 ta lai d˘t u = cos(lnx), dv = dx. Khi d´ du = . a . o 1− sin(lnx)dx, v = x v` a x I1 = x cos(lnx) + sin(lnx)dx. Thay I1 v`o biˆu th´.c dˆi v´.i I thu du.o.c phu.o.ng tr` a e’ u o o´ . ınh I = x(sin lnx − cos lnx) − Iv` t`. d´ a u o x I= (sin lnx − cos lnx) + C. 2 Nhˆn x´t. Trong c´c v´ du trˆn dˆy ta d˜ thˆy r˘ng t`. vi phˆn d˜ a e . a ı . e a a a ` ´ a u a a ´biˆt dv h`m v(x) x´c dinh khˆng do e a a . o .n tri. Tuy nhiˆn trong cˆng th´.c e o u . a ’ o e . a a ´(10.4) v` (10.4*) ta c´ thˆ chon v l` h`m bˆt k` v´ a y o .i vi phˆn d˜ cho a adv.
  • 27. 26 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . V´ du 6. T´ ı . ınh xdx dx 1) I = ; 2) In = , n ∈ N. sin2 x (x2 + a2)n Giai. 1) R˜ r`ng t´ phˆn n`y khˆng thuˆc bˆt c´. nh´m n`o ’ o a ıch a a o ´ o a u . o a dx trong ba nh´m d˜ nˆu. Thˆ nhu.ng b˘ng c´ch d˘t u = x, dv = o a e ´ e ` a a a . sin2 x v` ´p dung cˆng th´.c t´ phˆn t`.ng phˆn ta c´ aa . o u ıch a u ` a o cos x I = −xcotgx + cotgxdx = −xcotgx + dx sin x d(sin x) = −xcotgx + = −xcotgx + ln| sin x| + C. sin x 2) T´ phˆn In du.o.c biˆu diˆn du.´.i dang ıch a . ’ e ˜ e o . 1 x2 + a2 − x2 1 dx x2 dx In = 2 2 + a2 )n dx = 2 2 + a2 )n−1 − a (x a (x (x2 + a2)n 1 1 2xdx = 2 In−1 − 2 x 2 · a 2a (x + a2)n Ta t´ t´ phˆn o. vˆ phai b˘ng phu.o.ng ph´p t´ch phˆn t`.ng ınh ıch a ’ e ´ ’ ` a a ı a u 2xdx d(x2 + a2) ` phˆn. D˘t u = x, dv = 2 a a . = 2 . Khi d´ du = dx, o (x + a2 )n (x + a2 )n 1 v=− v` a (n − 1)(x2 + a2)n−1 1 2xdx −x 1 x = 2 + 2 In−1 2a2 (x2+a 2 )n 2a (n − 1)(x2 + a2 )n−1 2a (n − 1) T`. d´ suy r˘ng u o ` a 1 x 1 In = I 2 n−1 + 2 2 + a2 )n−1 − 2 In−1 a 2a (n − 1)(x 2a (n − 1) hay l` a x 2n − 3 In = + 2 In−1 . (*) 2a2 (n − 1)(x2 + a2 )n−1 2a (n − 1)
  • 28. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 27 Ta nhˆn x´t r˘ng t´ phˆn In khˆng thuˆc bˆt c´. nh´m n`o trong a e ` . a ıch a o . ´ o a u o a a ’ba nh´m d˜ chı ra. o Khi n = 1 ta c´o dx 1 x I1 = = arctg + C. x2 + a2 a a Ap dung cˆng th´.c truy hˆi (*) ta c´ thˆ t´nh I2 qua I1 rˆi I3 qua ´ . o u ` o ’ o e ı ` oI2,...V´ du 7. T´ I = ı . ınh xeax cos bxdx. Giai. D˘t u = x, dv = eax cos bxdx. Khi d´ du = dx, ’ a . o a cos bx + b sin bx v = eax a2 + b2(xem v´ du 4). Nhu. vˆy ı . a . a cos bx + b sin bx 1 I = xeax 2 + b2 − 2 eax(a cos bx + b sin bx)dx a a + b2 a cos bx + b sin bx a = xeax 2 + b2 − 2 eax cos bxdx a a + b2 b − 2 eax sin bxdx. a + b2 T´ phˆn th´. nhˆt o. vˆ phai du.o.c t´nh trong v´ du 4, t´ phˆn ıch a u ´ a ’ e ’ ´ . ı ı . ıch ath´. hai du.o.c t´ tu.o.ng tu. v` b˘ng u . ınh . a ` a a sin bx − b cos bx eax sin bxdx = eax · a2 + b2 Thay c´c kˆt qua thu du.o.c v`o biˆu th´.c dˆi v´.i I ta c´ a e ´ ’ . a ’ e u o o´ o eax a I= x− (a cos bx + b sin bx) a2 + b2 a2 + b2 b − (a sin bx − b cos bx) + C a2 + b2 ` ˆ BAI TAP .
  • 29. 28 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 2x (x ln 2 − 1) 1. x2x dx. (DS. ) ln2 2 2. x2 e−x dx. (DS. −x2e−x − 2xe−x − 2e−x ) 2 1 2 3. x3 e−x dx. (DS. − (x2 + 1)e−x ) 2 1 5x 3 3 2 31 31 4. (x3 + x)e5xdx. (DS. e x − x + x− ) 5 5 25 125 √ 5. arc sin xdx. (DS. xarc sin x + 1 − x2 ) 1 1 √ 6. xarc sin xdx. (DS. (2x2 − 1)arc sin x + x 1 − x2) 4 4 x3 2x2 + 1 √ 7. x2 arc sin 2xdx. (DS. arc sin 2x + 1 − 4x2) 3 36 1 8. arctgxdx. (DS. xarctgx − ln(1 + x2)) 2 √ √ √ 9. arctg xdx. (DS. (1 + x)arctg x − x) x4 − 1 x3 x 10. x3 arctgxdx. (DS. arctgx − + ) 4 12 4 x2 + 1 1 11. (arctgx)2xdx. (DS. (arctgx)2 − xarctgx + ln(1 + x2)) 2 2 √ 12. (arc sin x)2dx. (DS. x(arc sin x)2 + 2arc sin x 1 − x2 − 2x) arc sin x √ √ 13. √ dx. (DS. 2 x + 1arc sin x + 4 1 − x) x+1 √ arc sin x arc sin x 1 + 1 − x2 14. dx. (DS. − − ln ) x2 x x xarctgx √ √ 15. √ dx. (DS. 1 + x2arcrgx − ln(x + 1 + x2)) 1 + x2
  • 30. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn a a ınh ıch a 29 √ arc sin x √ √ √16. √ dx. (DS. 2( x − 1 − xarc sin x)) 1−x17. ln xdx. (DS. x(ln x − 1)) √ 2 3/2 4 818. x ln2 xdx. x (DS. ln2 x − ln x + ) 3 3 9 √ √ √19. ln(x + 16 + x2)dsx. (DS. x ln(x + 16 + x2) − 16 + x2 ) √ x ln(x + 1 + x2) √ √20. √ dx. (DS. 1 + x2 ln(x + 1 + x2) − x) 1 + x2 x21. sin x ln(tgx)dx. (DS. ln tg − cos x ln(tgx)) 2 (x3 + 1) ln(x + 1) x3 x2 x22. x2 ln(1 + x)dx. (DS. − + − ) 3 9 6 3 1 − 2x2 x23. x2 sin 2xdx. (DS. cos 2x + sin 2x) 4 2 124. x3 cos(2x2)dx. (DS. (2x2 sin 2x2 + cos 2x2)) 8 ex (sin x − cos x)25. ex sin xdx. (DS. ) 2 sin x + (ln 3) cos x x26. 3x cos xdx. (DS. 3 ) 1 + ln2 3 e3x27. e3x(sin 2x − cos 2x)dx. (DS. (sin 2x − 5 cos 2x)) 1328. xe2x sin 5xdx. e2x 21 20 (DS. 2x + sin 5x + − 5x + cos 5x ) 29 29 29 1 229. x2ex sin xdx. (DS. (x − 1) sin x − (x − 1)2 cos x ex) 2
  • 31. 30 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 2 x (x − 1)2 sin x + (x2 − 1) cos x x 30. x e cos xdx. (DS. e ) 2 [3 sin x(ln x) − cos(ln x)]x3 31. x2 sin(ln x)dx. (DS. ) 10 32. T` cˆng th´.c truy hˆi dˆi v´.i mˆ i t´ phˆn In du.o.c cho du.´.i ım o u ` o o o ´ ˜ o ıch a . o dˆy: a 1 n ax n 1) In = xn eaxdx, a = 0. (DS. In = x e − In−1 ) a a 2) In = lnn xdx. (DS. In = x lnn x − nIn−1 ) xα+1 lnn x n 3) In = xα lnn xdx, α = −1. (DS. In = − In−1 ) α+1 α+1 √ xn dx xn−1 x2 + a n − 1 4) In = √ , n > 2. (DS. In = − aIn−2 ) x2 + a n n n cos x sinn−1 x n − 1 5) In = sin xdx, n > 2. (DS. In = − + In−2 ) n n sin x cosn−1 x n − 1 6) In = cosn xdx, n > 2. (DS. In = + In−2 ) n n dx sin x n−2 7) In = nx , n > 2. (DS. In = n−1 x + In−2 ) cos (n − 1) cos n−1 10.2 C´c l´.p h`m kha t´ trong l´.p c´c a o a ’ ıch o a h`m so. cˆp a a´ 10.2.1 T´ phˆn c´c h`m h˜.u ty ıch a a a u ’ 1) Phu.o.ng ph´p hˆ sˆ bˆt dinh. H`m dang . ´ ´ a e o a . a . Pm (x) R(x) = Qn (x)
  • 32. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 31trong d´ Pm (x) l` da th´.c bˆc m, Qn (x) l` da th´.c bˆc n du.o.c goi l` o a u a . a u a . . . ah`m h˜ a u.u ty (hay phˆn th´.c h˜.u ty). Nˆu m ’ a u u ’ ´ e n th` Pm (x)/Qn (x) ıdu.o.c goi l` phˆn th´.c h˜.u ty khˆng thu.c su.; nˆu m < n th` . . a a u u ’ o . . ´ e ıPm (x)/Qn (x) du ..o.c goi l` phˆn th´.c h˜.u ty thu.c su.. u u ’ . . . a a ´ Nˆu R(x) l` phˆn th´ u ’ e a a u.c h˜.u ty khˆng thu.c su. th` nh`. ph´p chia o . . ı o eda th´.c ta c´ thˆ t´ch phˆn nguyˆn W (x) l` da th´.c sao cho u ’ o e a `a e a u Pm (x) Pk (x) R(x) = = W (x) + (10.5) Qn (x) Qn (x)trong d´ k < n v` W (x) l` da th´.c bˆc m − n. o a a u a. T`u. (10.5) suy r˘ng viˆc t´ t´ch phˆn phˆn th´.c h˜.u ty khˆng ` a e ınh ı a a u u ’ o . .c su. du.o.c quy vˆ t´nh t´ phˆn phˆn th´.c h˜.u ty thu.c su. v` t´chthu . ` ı e ıch a a u u ’ . . a ı . .phˆn mˆt da th´ a o u.c. .Dinh l´ 10.2.1. Gia su. Pm (x)/Qn (x) l` phˆn th´.c h˜.u ty thu.c su.-. y ’ ’ a a u u ’ . .v` a Q(x) = (x − a)α · · · (x − b)β (x2 + px + q)γ · · · (x2 + rx + s)δtrong d´ a, . . . , b l` c´c nghiˆm thu.c, x2 + px + q, . . . , x2 + rx + s l` o a a e . . anh˜.ng tam th´.c bˆc hai khˆng c´ nghiˆm thu.c. Khi d´ u u a . o o e . . oP (x) Aα A1 Bβ Bβ−1 = + ··· + + ··· + + + ···+Q(x) (x − a)α x−a (x − b)β (x − b)β−1 B1 Mγ x + Nγ M1 x + N1 + + 2 + ··· + 2 + ···+ x − b (x + px + q)γ x + px + q Kδ x + Lδ K1 x + L1 + 2 δ + ··· + 2 , (10.6) (x + rx + s) x + rx + strong d´ Ai, Bi , Mi , Ni , Ki v` Li l` c´c sˆ thu.c. o a a a o . ´ a a u.c o. vˆ phai cua (10.6) du.o.c goi l` c´c phˆn th´.c do.n C´c phˆn th´ ’ e ’ ’´ . . a a a ugian hay c´c phˆn th´.c co. ban v` d˘ng th´.c (10.6) du.o.c goi l` khai ’ a a u ’ a a ’ u . . a ’ u.c h˜.u ty thu.c su. P (x)/Q(x) th`nh tˆng c´c phˆn th´.ctriˆn phˆn th´ u ’ . . e a a o’ a a u . ban v´.i hˆ sˆ thu.c.co ’ . ´ o e o . ’ . ´ ’ Dˆ t´ c´c hˆ sˆ Ai , Bi , . . . , Ki , Li ta c´ thˆ ´p dung e ınh a e o o ea .
  • 33. 32 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . Phu.o.ng ph´p I. Quy dˆng mˆ u sˆ d˘ng th´.c (10.6) v` sau d´ cˆn a ` o ˜ ´ ’ a o a u a o a ` .a c`ng bˆc cua biˆn x v` di dˆn hˆ phu.o.ng ´ ´ . . ´ a e o ’ u b˘ng c´c hˆ sˆ cua l˜y th` u a u a ’ . e a e e tr` dˆ x´c dinh Ai , . . . , Li (phu.o.ng ph´p hˆ sˆ bˆt dinh). ’ ınh e a . . ´ ´ a e o a . .o.ng ph´p II. C´c hˆ sˆ Ai , . . . , Li c˜ng c´ thˆ x´c dinh b˘ng ’ ` Phu a a e o . ´ u o e a . a .c tu.o.ng du.o.ng v´.i (10.6)) bo.i a . ’ c´ch thay x trong (10.6) (ho˘c d˘ng th´ a a u o ’ c´c sˆ du.o.c chon mˆt c´ch th´ch ho.p. a o ´ . . o a . ı . T` u. (10.6) ta c´o D.nh l´ 10.2.2. T´ phˆn bˆt dinh cua moi h`m h˜.u ty dˆu biˆu -i y ıch a a . ´ ’ . a u ’ ` e ’ e diˆn du.o.c qua c´c h`m so. cˆp m` cu thˆ l` qua c´c h`m h˜.u ty, h`m ˜e . a a ´ a a . e a ’ a a u ’ a lˆgarit v` h`m arctang. o a a CAC V´ DU ´ I . xdx V´ du 1. T´ I = ı . ınh (x − 1)(x + 1)2 ’ Giai. Ta c´ o x A B1 B2 2 = + + (x − 1)(x + 1) x − 1 x + 1 (x + 1)2 T`. d´ suy r˘ng u o ` a x = A(x + 1)2 + B1(x − 1)(x + 1) + B2 (x − 1). (10.7) Ta x´c dinh c´c hˆ sˆ A, B1 , B2 b˘ng c´c phu.o.ng ph´p sau dˆy. a . . ´ a e o ` a a a a Phu.o.ng ph´p I. Viˆt d˘ng th´.c (10.7) du.´.i dang a ´ ’ e a u o . x ≡ (A + B1 )x2 + (2A + B2)x + (A − B1 − B2 ). Cˆn b˘ng c´c hˆ sˆ cua l˜y th`.a c`ng bˆc cua x ta thu du.o.c a a ` . ´ a e o ’ u u u a ’ . .   A + B1 = 0  2A + B2 = 1   A − B1 − B2 = 0. 1 1 1 T`. d´ A = , B1 = − , B2 = . u o 4 4 2
  • 34. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 33 1 Phu.o.ng ph´p II. Thay x = 1 v`o (10.7) ta c´ 1 = A · 4 ⇒ A = . a a o 4Tiˆp theo, thay x = −1 v`o (10.7) ta thu du.o.c: −1 = −B2 · 2 hay ´ e a . 1l` B2 = . Dˆ t`m B1 ta thˆ gi´ tri x = 0 v`o (10.7) v` thu du.o.c a ’ e ı ´ e a . a a . 2 10 = A − B1 − B2 hay l` B1 = A − B2 = − . a 4 Do d´o 1 dx 1 dx 1 dx I= − + 4 x−1 4 x+1 2 (x + 1)2 1 1 x−1 =− + ln + C. 2(x + 1) 4 x+1 3x + 1V´ du 2. T´ I = ı . ınh dx. x(1 + x2)2 Giai. Khai triˆn h`m du.´.i dˆu t´ch phˆn th`nh tˆng c´c phˆn ’ ’ e a o a ı´ a a ’ o a ath´.c co. ban u ’ 3x + 1 A Bx + C Dx + F 2 )2 = + 2 + x(1 + x x 1+x (1 + x2 )2T`. d´ u o 3x + 1 ≡ (A + B)x4 + Cx3 + (2A + B + D)x2 + (C + F )x + A.Cˆn b˘ng c´c hˆ sˆ cua c´c l˜y th`.a c`ng bˆc cua x ta thu du.o.c a ` a . ´ a e o ’ a u u u a ’ . .   A+B =0     C =0   2A + B + D = 0 ⇒ A = 1, B = −1, C = 0, D = −1, F = 3    C +F =3     A = 1.T`. d´ suy r˘ng u o ` a dx xdx xdx dx I= − − +3 x 1 + x2 (1 + x2)2 (1 + x2)2 1 1 dx = ln |x| − ln(1 + x2 ) − (1 + x2 )−2 d(1 + x2) + 3 2 2 (1 + x2 )2 1 1 = ln |x| − ln(1 + x2 ) + + 3I2 . 2 2(1 + x2 )
  • 35. 34 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . dx Ta t´ I2 = ınh b˘ng cˆng th´.c truy hˆi thu du.o.c trong ` a o u ` o . (1 + x2 )2 10.1. Ta c´ o 1 x 1 x 1 dx I2 = · 2 + I1 = 2) + 2 1+x 2 2(1 + x 2 1 + x2 x 1 = 2) + arctgx + C. 2(1 + x 2 Cuˆi c`ng ta thu du.o.c ´ o u . 1 3x + 1 3 I = ln |x| − ln(1 + x2 ) + + arctgx + C. 2 2(1 + x2) 2 ` ˆ BAI TAP . T´ c´c t´ phˆn (1-12) ınh a ıch a xdx 1. . (x + 1)(x + 2)(x − 3) 1 2 3 (DS. ln |x + 1| − ln |x + 2| + |x − 3|) 4 5 20 2x4 + 5x2 − 2 2. dx. 2x3 − x − 1 x2 DS. + ln |x − 1| + ln(2x2 + 2x + 1) + arctg(2x + 1)) 2 2x3 + x2 + 5x + 1 3. dx. (x2 + 3)(x2 − x + 1) 1 x 2 2x − 1 DS. √ arctg √ + ln(x2 − x + 1) + √ arctg √ ) 3 3 3 3 x4 + x2 + 1 4. dx. x(x − 2)(x + 2) x2 1 21 21 (DS. − ln |x| + ln |x − 2| + ln |x + 2|) 2 4 8 8
  • 36. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 35 dx5. . x(x − 1)(x2 − x + 1)2 x−1 10 2x − 1 1 2x − 1 (DS. ln − √ arctg √ − ) x 3 3 3 3 x2 − x + 1 x4 − x2 + 16. dx. (x2 − 1)(x2 + 4)(x2 − 2) √ 1 x−1 7 x 1 x− 2 (DS. − ln + arctg + √ ln √ ) 10 x+1 20 2 4 2 x+ 2 3x2 + 5x + 127. dx. (x2 + 3)(x2 + 1) √ 5 2 3 x 5 9 (DS. − ln(x + 3) − arctg √ + ln(x2 + 1) + arctgx) 4 2 3 4 2 (x4 + 1)dx8. . x5 + x4 − x3 − x2 1 1 1 1 (DS. ln |x| + + ln |x − 1| − ln |x + 1| + ) x 2 2 x+1 x3 + x + 19. dx. x4 − 1 3 1 1 (DS. ln |x − 1| + ln |x + 1| − arctgx) 4 4 2 x410. dx. 1 − x4 x+1 1 (DS. − x + ln + arctgx) x−1 2 3x + 511. dx. (x2 + 2x + 2)2 2x − 1 (DS. + arctg(x + 1)) 2(x2+ 2x + 2)
  • 37. 36 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . x4 − 2x2 + 2 12. dx. (x2 − 2x + 2)2 3−x (DS. x + + 2 ln(x2 − 2x + 2) + arctg(x − 1)) x2 − 2x + 2 x2 + 2x + 7 13. dx. (x − 2)(x2 + 1)3 3 3 1−x 11 (DS. ln |x2 − 2| − ln |x2 + 1| + 2 − arctgx) 5 10 x +1 5 x2 14. dx. (x + 2)2 (x + 1) 4 (DS. + ln |x + 1|) x+2 x2 + 1 15. dx. (x − 1)3 (x + 3) 1 3 5 x−1 (DS. − 2 − + ln ) 4(x − 1) 8(x − 1) 32 x+3 dx 16. x5 − x2 1 1 (x − 1)2 1 2x + 1 (DS. + ln 2 + √ arctg √ ) x 6 x +x+1 3 3 3x2 + 8 17. dx. x3 + 4x2 + 4x 10 (DS. 2 ln |x| + ln |x + 2| + ) x+2 2x5 + 6x3 + 1 18. dx. x4 + 3x2 1 1 x (DS. x2 − − √ arctg √ ) 3x 3 3 3
  • 38. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 37 x3 + 4x2 − 2x + 119. dx. x4 + x |x|(x2 − x + 1) 2 2x − 1 (DS. ln 2 + √ arctg √ ) (x + 1) 3 3 x3 − 320. dx. x4 + 10x2 + 25 1 25 − 3x 3 x (DS. ln(x2 + 5) + 2 + 5) − √ arctg √ ) 2 10(x 10 5 5 ’ ˜ Chı dˆ n. x4 + 10x2 + 25 = (x2 + 5)2 . a10.2.2 T´ phˆn mˆt sˆ h`m vˆ ty do.n gian ıch a ´ o o a . o ’ ’Mˆt sˆ t´ phˆn h`m vˆ ty thu.`.ng g˘p c´ thˆ t´nh du.o.c b˘ng phu.o.ng . ´ o o ıch a a o ’ o a o e ı . ’ . ` a a u .u ty h´a h`m du.´.i dˆu t´ch phˆn. Nˆi dung cua phu.o.ng ph´pph´p h˜ ’ o a o a ı´ a o ’ a . a a ım o e ´ ’n`y l` t` mˆt ph´p biˆn dˆi du ı e o .a t´ch phˆn d˜ cho cua h`m vˆ ty vˆ a a ’ a o ’ ` e .t´ phˆn h`m h˜.u ty. Trong tiˆt n`y ta tr` b`y nh˜.ng ph´p dˆi ıch a a u ’ ´ e a ınh a u e o ’biˆn cho ph´p h˜.u ty h´a dˆi v´.i mˆt sˆ l´.p h`m vˆ ty quan trong ´ e e u ’ o o o ´ . ´ o o o a o ’ . ´ a .´.c k´ hiˆu R(x1 , x2, . . . ) hay r(x1 , x2, . . . ) l` h`m h˜.unhˆt. Ta quy u o y e a a u .ty dˆi v´.i mˆ i biˆn x1, x2 , . . . , xn . ’ o o´ ˜ e o ´ ıch a a a o ’ a I. T´ phˆn c´c h`m vˆ ty phˆn tuyˆn t´ ´ e ınh. T´ phˆn dang ıch a . ax + b p1 ax + b pn R x, ,..., dx (10.8) cx + d cx + dtrong d´ n ∈ N; p1 , . . . , pn ∈ Q; a, b, c ∈ R; ad − bc = 0 du.o.c h˜.u ty o . u ’h´a nh`. ph´p dˆi biˆn o o e o e ’ ´ ax + b = tm cx + do. dˆy m l` mˆ u sˆ chung cua c´c sˆ h˜.u ty p1 , . . . , pn .’ a a ˜ o a ´ ´ ’ a o u ’ II. T´ phˆn dang ıch a . √ R(x, ax2 + bx + c)dx, a = 0, b2 − 4ac = 0 (10.9)
  • 39. 38 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . c´ thˆ h˜.u ty h´a nh`. ph´p thˆ Euler: ’ o e u ’ o o e ´ e √ √ ´ (i) ax2 + bx + c = ± ax ± t, nˆu a > 0; e √ √ ´ (ii) ax2 + bx + c = ±xt ± c, nˆu c > 0; e √ (iii) ax2 + bx + c = ±(x − x1 )t √ ax2 + bx + c = ±(x − x2 )t trong d´ x1 v` x2 l` c´c nghiˆm thu.c kh´c nhau cua tam th´.c bˆc hai o a a a e . . a ’ u a . 2 ´ a ’ ax + nbx + c. (Dˆu o a e . c´c vˆ phai cua d˘ng th´.c c´ thˆ lˆy theo tˆ ´ ’ ’ a ’ u o e a ’ ´ o’ ho.p t`y y). . u ´ III. T´ phˆn cua vi phˆn nhi th´.c. D´ l` nh˜.ng t´ phˆn dang ıch a ’ a . u o a u ıch a . xm (axn + b)pdx (10.10) trong d´ a, b ∈ R, m, n, p ∈ Q v` a = 0, b = 0, n = 0, p = 0; biˆu th´.c o a e’ u m n p x (zx + b) du . .o.c goi l` vi phˆn nhi th´.c. . a a . u T´ phˆn vi phˆn nhi th´.c (10.10) du.a du.o.c vˆ t´ch phˆn h`m ıch a a . u . ` ı e a a u.u ty trong ba tru.`.ng ho.p sau dˆy: h˜ ’ o a . a o´ 1) p l` sˆ nguyˆn, e m+1 2) a o´ l` sˆ nguyˆn, e n m+1 3) ´ + p l` sˆ nguyˆn. a o e n Dinh l´ (Trebu.s´p). T´ phˆn vi phˆn nhi th´.c (10.10) biˆu diˆn -. y e ıch a a . u ’ e ˜ e du.o.c du.´.i dang h˜.u han nh`. c´c h`m so. cˆp (t´.c l` du.a du.o.c vˆ . o . u . o a a ´ a u a . ` e t´ch phˆn h`m h˜ ’ ı a a u.u ty hay h˜.u ty h´a du.o.c) khi v` chı khi ´ nhˆt mˆt u ’ o a ’ ´ . ıt a o . m+1 m+1 ´ trong ba sˆ p, o , ´ + p l` sˆ nguyˆn. a o e n n 1) Nˆu p l` sˆ nguyˆn th` ph´p h˜.u ty h´a s˜ l` ´ e a o ´ e ı e u ’ o e a x = tN trong d´ N l` mˆ u sˆ chung cua c´c phˆn th´.c m v` n. o a ˜ o a ´ ’ a a u a m+1 ´ 2) Nˆu e ´ l` sˆ nguyˆn th` d˘t a o e ı a. n axn + b = tM
  • 40. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 39 a ˜ o ’ a ´trong d´ M l` mˆ u sˆ cua p. o m+1 ´ 3) Nˆu e ´ + p l` sˆ nguyˆn th` d˘t a o e ı a. n a + bx−n = tM a ˜ o ’ a ´trong d´ M l` mˆ u sˆ cua p. o CAC V´ DU ´ I .V´ du 1. T´ ı . ınh √ 3 √ x + x2 + 6 x dx 1) I1 = √ dx , 2) I2 = · x(1 + 3 x) 3 (2 + x)(2 − x)5 1 ’ Giai. 1) T´ phˆn d˜ cho c´ dang I, trong d´ p1 = 1, p2 = , ıch a a o . o 3 1 ˜ ´p3 = . Mˆ u sˆ chung cua p1 , p2 , p3 l` m = 6. Do d´ ta d˘t x = t6. a o ’ a o a . 6Khi d´: o t 6 + t4 + t 5 t 5 + t3 + 1 I=6 t dt = 6 dt t6(1 + t2) 1 + t2 dt 3√ 2 3 √ =6 t3dt + 6 2 = x + 6arctg 6 x + C. 1+t 2 2) B˘ng ph´p biˆn dˆi so. cˆp ta c´ ` a e ´ ’ e o ´ a o 3 2 − x dx I2 = · 2 + x (2 − x)2D´ l` t´ phˆn dang I. Ta d˘t o a ıch a . a . 2−x = t3 2+xv` thu du.o.c a . 1 − t3 t2dt x=2 , dx = −12 · 1 + t3 (1 + t3)2
  • 41. 40 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . T`. d´ u o t3 (t3 + 1)2 dt 3 dt 3 3 2+x 2 I2 = −12 6 (t3 + 1)2 =− 3 = + C. 16t 4 t 8 2−x V´ du 2. T´ c´c t´ phˆn ı . ınh a ıch a dx dx 1) I1 = √ , 2) I2 = √ , x x2 + x + 1 (x − 2) −x2 + 4x − 3 dx 3) I3 = √ ,· (x + 1) 1 + x − x2 Giai. 1) T´ phˆn I1 l` t´ch phˆn dang II v` a = 1 > 0 nˆn ta su. ’ ıch a a ı a . a e ’ . e e´ dung ph´p thˆ Euler (i) √ x2 + x + 1 = x + t, x2 + x + 1 = x2 + 2tx + t2 t2 − 1 √ −t2 + t − 1 x= , x2 + x + 1 = x + t = 1 − 2t 1 − 2t 2 2(−t + t − 1) dx = dt. (1 − 2t)2 T`. d´ u o √ dt 1−t 1 + x − x2 + x + 1 I1 = 2 = ln + C = ln √ + C. t2 − 1 1+t 1 − x + x2 + x + 1 2) Dˆi v´.i t´ phˆn I2 (dang II) ta c´ ´ o o ıch a . o −x2 + 4x − 3 = −(x − 1)(x − 3) v` do d´ ta su. dung ph´p thˆ Euler (iii): a o ’ . e ´ e √ −x2 + 4x − 3 = t(x − 1). o Khi d´ 3−x −(x − 1)(x − 3) = t2(x − 1)2 , −(x − 3) = t2 (x − 1), t= , x−1 t2 + 3 √ 2t x= 2+1 , −x2 + 4x − 3 = t(x − 1) = 2 t t +1 −4tdt dx = 2 (t + 1)2
  • 42. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 41v` thu du.o.c a . √ √ dt 1−t x−1− 3−x I2 = 2 = ln + C = ln √ √ + C. t2 − 1 1+t x−1+ 3−x 3) Dˆi v´.i t´ phˆn I3 (dang III) ta c´ C = 1 > 0. Ta su. dung ´ o o ıch a . o ’ . e ´ph´p thˆ Euler (ii) v` e a √ 1 + x − x2 = tx − 1, 1 + x − x2 = t2x2 − 2tx + 1, 2t + 1 √ t2 + t − 1 x= 2 , 1 + x − x2 = tx − 1 = , t +√ 1 t2 + 1 1 + 1 + x − x2 −2(t2 + t − 1) t= , dx = · x (t2 + 1)2 oDo d´ dt d(t + 1) I3 = −2 = −2 = −2arctg(t + 1) + C t2 + 2t + 2 1 + (t + 1)2 √ 1 + x + 1 + x − x2 = −2arctg + C. xV´ du 3. T´ c´c t´ phˆn ı . ınh a ıch a √ x √ 4 11) I1 = √ 2 dx, x 0; 2) I2 = x 1−√ dx; (1 + x) 3 x3 dx3) I3 = · x2 3 (1 + x3)5 ’ Giai. 1) Ta c´ o 1 1 −2 I1 = x2 1 + x3 dx, 1 1trong d´ m = o ˜ ´ ’ ` , n = , p = −2, mˆ u sˆ chung cua m v` n b˘ng 6. a o a a 2 3V` p = −2 l` sˆ nguyˆn, ta ´p dung ph´p dˆi biˆn x = t6 v` thu du.o.c ı a o´ e a . e o e’ ´ a . t8 4t2 + 3 I1 = 6 dt = 6 t4 − 2t2 + 3 − dt (1 + t2)2 (1 + t2)2 6 dt t2 = t5 − 4t3 + 18t − 18 −6 dt. 5 1 + t2 (1 + t2 )2
  • 43. 42 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . V` ı t2 dt 1 1 t 1 2 )2 =− td =− + arctgt (1 + t 2 1 + t2 2(1 + t2) 2 nˆn cuˆi c`ng ta thu du.o.c e ´ o u . 6 3x1/6 I1 = x5/6 − 4x1/2 + 18x1/6 + − 21arctgx1/6 + C. 5 1 + x1/3 2) Ta viˆt I2 du.´.i dang ´ e o . 1 3 1 I2 = x 2 1 − x− 2 4 dx. ’. a 1 3 1 m+1 O dˆy m = , n = − , p = v` a ´ = −1 l` sˆ nguyˆn v` ta a o e a 2 2 4 n c´ tru.`.ng ho.p th´. hai. Ta su. dung ph´p dˆi biˆn o o . u ’ . ’ ´ e o e 1 1 − √ = t4 . x3 2 8 5 Khi d´ x = (1 − t4)− 3 , dx = (1 − t4)− 3 t3dt v` do vˆy o a a . 3 8 t4 2 1 2 t dt I2 = 4 )2 dt = td 4 = − 3 (1 − t 3 1−t 3 1 − t4 1 − t2 2t 1 1 1 = 4) − 2 + dt 3(1 − t 3 1−t 1 + t2 2t 1 1+t 1 = − ln − arctgt + C, 3(1 − t4 ) 6 1−t 3 1/4 trong d´ t = 1 − x−3/2 o . ´ .´.i dang 3) Ta viˆt I3 du o . e 5 I3 = x−2 (1 + x3 )− 3 dx. ’. a 5 m+1 O dˆy m = −2, n = 3, p = − v` a ´ + p = −2 l` sˆ nguyˆn. a o e 3 n Do vˆy ta c´ tru.`.ng ho.p th´. ba. Ta thu.c hiˆn ph´p dˆi biˆn a . o o . u . e . e o e’ ´ 1 + x−3 = t3 ⇒ 1 + x3 = t3x3 .
  • 44. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 43T`. d´ u o 1 t3 1 x3 = 3−1 , 1 + x3 = 3 , x = (t3 − 1)− 3 t t −1 −4 2 dx = −t (t − 1) 3 dt, x−2 = (t3 − 1) 3 . 2 3Do vˆy a . t3 −5/3 4 1 − t3 I3 = − (t3 − 1)2/3 3−1 t2 (t3 − 1)− 3 dt = dt t t3 −3 t−2 1 + 2t3 = t dt − dt = −t+C = C − −2 2t3 3 2 + 3x =C− · 2x (1 + x3 )2 3
  • 45. 44 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . ` ˆ BAI TAP . T´ c´c t´ phˆn (1-15) ınh a ıch a dx 1. √ √ . 2x − 1 − 3 2x − 1 3 (DS. u3 + u2 + 3u + 3 ln |u − 1|, u6 = 2x − 1) 2 xdx 2. √ . (3x − 1) 3x − 1 2 3x − 2 (DS. √ ) 9 3x − 1 1 − x dx 3. . 1+x x √ 1− 1 − x2 (DS. − arc sin x) x 3 x + 1 dx 4. . x−1 x+1 1 (1 − t)2 √ 2t + 1 3 x+1 (DS. − ln 2 + 3arctg √ , t= ) 2 1+t+t 3 x−1 √ √ x+1− x−1 5. √ √ dx. x+1+ x−1 1 2 √ √ (DS. (x − x x2 − 1 + ln |x + x2 − 1|) 2 xdx 6. √ √ . x+1− 3x+1 1 9 1 8 1 7 1 6 1 5 1 4 (DS. 6 u + u + u + u + u + u , u6 = x + 1) 9 8 7 6 5 4
  • 46. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 45 1+x7. (x − 2) dx. 1−x 1 √ 3 (DS. 1 − x 1 − x2 − arc sin x) 2 2 3 x+1 dx8. . x − 1 (x − 1)3 3 3 x+1 4 3 3 x+1 3 (DS. − ) 16 x−1 28 x−1 dx x−29. . (DS. 2 ) (x − 1)3 (x − 2) x−1 x−1 ’ a˜ ´ Chı dˆ n. Viˆt e (x − 1)3 (x − 2) = (x − 1)(x − 2) , d˘t a x−2 . x−1t= . x−2 dx10. . 3 (x − 1)2 (x + 1) 1 u2 + u + 1 √ 2u + 1 x+1 (DS. ln 2 − 3arctg √ , u3 = ) 2 u − 2u + 1 3 x−1 dx 3 3 1+x11. . (DS. ) 3 (x + 1)2 (x − 1)4 2 x−1 dx 4 4 x−112. . (DS. ) 4 (x − 1)3 (x + 2)5 3 x+2 dx 3 3x − 5 3 x+113. . (DS. ) 3 (x − 1)7 (x + 1)2 16 x − 1 x−1 dx x−514. . (DS. −3 6 ) 6 (x − 7)7 (x − 5)5 x−7 dx n n x−b15. , a = b. (DS. ) n (x − a)n+1 (x − b)n−1 b−a x−a
  • 47. 46 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . √ √ x+1− x−1 16. √ √ dx. x+1+ x−1 √ x 2 x x2 − 1 1 √ (DS. − + ln |x + x2 − 1|) 3 2 2 Su. dung c´c ph´p thˆ Euler dˆ t´nh c´c t´ch phˆn sau dˆy (17-22) ’ . a e ´ e ’ e ı a ı a a √ dx 1 + x − x2 + x + 1 17. √ . (DS. ln √ ) x x2 + x + 1 1 − x + x2 + x + 1 √ √ dx x−1− 3−x 18. √ . (DS. ln √ √ ) (x − 2) −x2 + 4x − 3 x−1+ 3−x √ dx 1 + x + 1 + x − x2 19. √ . (DS. −2arctg ) (x + 1) 1 + x − x2 x dx 20. √ . (x − 1) x2 + x + 1 √ 3 x−1 (DS. ln ) 3 3 + 3x + 2 3(x2 + x + 1) (x − 1)dx 1 + 2x 21. √ . (DS. √ ) (x2 + 2x) x2 + 2x x2 + 2x 5x + 4 22. √ dx. x2 + 2x + 5 √ √ (DS. 5 x2 + 2x + 5 − ln x + 1 + x2 + 2x + 5 ) 1 ˜ ’ ’ ´ Chı dˆ n. C´ thˆ dˆi biˆn t = (x2 + 2x + 5) = x + 1. ’ a o e o e 2 T´ c´c t´ phˆn cua vi phˆn nhi th´.c ınh a ıch a ’ a . u 1 1 1 1 1 23. x− 3 (1 − x1/6)−1 dx. (DS. 6x 6 + 3x 3 + 2x 2 + 6 ln x 6 − 1 ) 2 1 3 1 24. x− 3 (1 + x 3 )−3 dx. (DS. − (1 + x 3 )−2 ) 2 1 1 4 1 1 1 25. x− 2 (1 + x 4 )−10 dx. (DS. (1 + x 4 )−9 − (1 + x 4 )−8 ) 9 2
  • 48. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 47 x t5 2t3 √26. √ dx. (DS. 3 − + t , t = 1 + x2/3) 1+ 3 x2 5 3 2 x2 + 127. x3(1 + 2x2)− 3 dx. (DS. √ ) 2 2x2 + 1 dx 1 √28. √ . (DS. x−3 (2x2 − 1) x2 + 1) x4 1 + x 2 3 dx 1 229. 2 (1 + x3 )5/3 . (DS. − x−1 (3x + 4)(2 + x3)− 3 ) x 8 dx 3 330. √ 3 √ . 4 (DS. −2 (x− 4 + 1)2 ) x3 1 + x3 dx 331. √ √ . (DS. − √ ) 3 x2( 3 x + 1)3 2( x + 1)2 3 √ 3 x32. √ dx. 3 x+1 u7 3 5 √ (DS. 6 − u + u3 − u2 , u2 = 3 x+1 ) 7 5 dx33. √ . x6 x2 − 1 u5 2u3 √ (DS. − + u, u= 1 − x−2 ) 5 3 dx34. √ 3 . x 1 + x5 √ 1 u2 − 2u + 1 3 2u + 1 (DS. ln 2 + arctg √ , u3 = 1 + x 5 ) 10 u +u+1 5 3 √35. x7 1 + x2 dx. u9 3u7 3u5 u3 (DS. − + − , u2 = 1 + x 2 ) 9 7 5 3
  • 49. 48 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . dx 36. √ 3 . 1 + x3 1 u2 + u + 1 1 2u + 1 (DS. ln 2 − √ arctg √ , u3 = 1 + x−3 ) 6 u − 2u + 1 3 3 dx 37. √ 4 . 1 + x4 1 u+1 1 (DS. ln − arctgu, u4 = 1 + x−4 ) 4 u−1 2 √ 3 38. x − x3dx. u 1 u2 + 2u + 1 1 2u − 1 (DS. − ln 2 − √ arctg √ , u3 = x−2 − 1) 2(u3 + 1) 12 u −u+1 2 3 3 10.2.3 T´ phˆn c´c h`m lu.o.ng gi´c ıch a a a . a I. T´ phˆn dang ıch a . R(sin x, cos x)dx (10.11) trong d´ R(u, v) l` h`m h˜.u ty cua c´c biˆn u b` v luˆn luˆn c´ thˆ o a a u ’ ’ a ´ e a o o o e ’ x h˜.u ty h´a du.o.c nh`. ph´p dˆi biˆn t = tg , x ∈ (−π, π). T`. d´ u ’ o . o e o e’ ´ u o 2 2t 1 − t2 2dt sin x = , cos x = , dx = · 1 + t2 1 + t2 1 + t2 Nhu.o.c diˆm cua ph´p h˜.u ty h´a n`y l` n´ thu.`.ng du.a dˆn nh˜.ng . ’ e ’ e u ’ o a a o o ´ e u ´ t´ to´n rˆt ph´ . ınh a a u.c tap. V` vˆy, trong nhiˆu tru.`.ng ho.p ph´p h˜.u ty h´a c´ thˆ thu.c hiˆn ı a. ` e o . e u ’ o o e . ’ e . .o.c nh`. nh˜.ng ph´p dˆi biˆn kh´c. du . o u ’ e e o ´ a II. Nˆu R(− sin x, cos x) = −R(sin x, cos x) th` su. dung ph´p dˆi e´ ı ’ . e o ’ ´ biˆn e t = cos x, x ∈ (0, π)
  • 50. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 49v` l´c d´ a u o dt dx = − √ 1 − t2 III. Nˆu R(sin x, − cos x) = −R(sin x, cos x) th` su. dung ph´p dˆi ´ e ı ’ . e o ’ ´biˆn e dt π π t = sin x, dx = √ , x∈ − , . 1 − t2 2 2 IV. Nˆu R(− sin x, − cos x) = R(sin x, cos x) th` ph´p h˜.u ty h´a ´ e ı e u ’ o π πs˜ l` t = tgx, x ∈ − , e a : 2 2 t 1 dt sin x = √ , cos x = √ , x = arctgt, dx = · 1 + t2 1 + t2 1 + t2 V. Tru.`.ng ho.p riˆng cua t´ phˆn dang (10.11) l` t´ch phˆn o . e ’ ıch a . a ı a sinm x cosn xdx, m, n ∈ Z (10.12) ´ ´ ’ ı a ´ ’ ı a (i) Nˆu sˆ m le th` d˘t t = cos x, nˆu n le th` d˘t sin x = t. e o . e . ´ (ii) Nˆu m v` n l` nh˜ e a a u .ng sˆ ch˘ n khˆng ˆm th` tˆt ho.n hˆt l` thay o ˜ ´ a o a ı o ´ ´ e asin x v` cos2 x theo c´c cˆng th´.c 2 a a o u 1 1 sin2 x = (1 − cos 2x), cos2 x = (1 + cos 2x). 2 2 ´ e a ˜ a o o o oa . ´ (iii) Nˆu m v` n ch˘ n, trong d´ c´ mˆt sˆ ˆm th` ph´p dˆi biˆn s˜ ı e o e e’ ´l` tgx = t hay cotgx = t. a (iv) Nˆu m + n = −2k, k ∈ N th` viˆt biˆu th´.c du.´.i dˆu t´ch ´ e ı e´ e ’ u ´ o a ıphˆn bo . a ’.i dang phˆn th´.c v` t´ch cos2 x (ho˘c sin2 x) ra khoi mˆ u sˆ. a u a a a ˜ ´ ’ a o . dx dxBiˆu th´.c ’ e u (ho˘c a . ) du.o.c thay bo.i d(tgx) (ho˘c d(cotgx)) . ’ a . cos2 x sin2 x . e o e ’ ´v` ´p dung ph´p dˆi biˆn t = tgx (ho˘c t = cotgx). aa a. VI. T´ phˆn dang ıch a . sinα x cosβ xdx, α, β ∈ Q. (10.13)
  • 51. 50 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . B˘ng ph´p dˆi biˆn sin2 x = t ta thu du.o.c ` a e o e ’ ´ . 1 α−1 β−1 I= t 2 (1 − t) 2 dt 2 v` b`i to´n du.o.c quy vˆ t´ch phˆn cua vi phˆn nhi th´.c. a a a . ` ı e a ’ a . u CAC V´ DU ´ I . V´ du 1. T´ t´ phˆn ı . ınh ıch a dx I= 3 sin x + 4 cos x + 5 x ’ Giai. D˘t t = tg , x ∈ (−π, π). Khi d´ a . o 2 dt I =2 = 2 (t + 3)−2 dt t2 + 6t + 9 2 2 =− +C =− x + C. t+3 3 + tg 2 V´ du 2. T´ ı . ınh dx J= (3 + cos 5x) sin 5x Giai. D˘t 5x = t. Ta thu du.o.c ’ a . . 1 dt J= 5 (3 + cos t) sin t v` (tru.`.ng ho.p II) do d´ b˘ng c´ch d˘t ph´p dˆi biˆn z = cos t ta c´ a o . o a` a a . e o e’ ´ o 1 dz 1 A B C J= 2 − 1) = + + dz 5 (z + 3)(z 5 z−1 z−1 z+3 1 1 1 1 = − + dz 5 8(z − 1) 4(z + 1) 8(z + 3) 1 1 1 1 = ln |z − 1| − ln |z + 1| + ln |z + 3| + C 5 8 4 8 1 (z − 1)(z + 3) = ln +C 40 (z + 1)2 1 cos2 x + 2 cos 5x − 3 = ln + C. 40 (cos 5x + 1)2
  • 52. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 51V´ du 3. T´ ı . ınh 2 sin x + 3 cos x J= dx sin2 x cos x + 9 cos3 x Giai. H`m du.´.i dˆu t´ phˆn c´ t´ chˆt l` ’ a ´ o a ıch a o ınh a a ´ R(− sin x, − cos x) = R(sin x, cos x). π πDo d´ ta su. dung ph´p dˆi biˆn t = tgx, x ∈ − , o ’ . e o e’ ´ . Chia tu. sˆ ’ o ´ 2 2v` mˆ u sˆ cua biˆu th´.c du.´.i dˆu t´ phˆn cho cos3 x ta c´ a ˜ o ’ a ´ ’ e u ´ o a ıch a o 2tgx + 3 2t + 3 J= 2 d(tgx) = dt tg x + 9 t2 + 9 t = ln(t2 + 9) + arctg +C 3 tgx = ln(tg2 x + 9) + arctg + C. 3V´ du 4. T´ ı . ınh dx J= 6 sin x + cos6 x Giai. Ap dung cˆng th´.c ’ ´ . o u 1 1 cos2 x = (1 + cos 2x), sin2 x = (1 − sin 2x) 2 2ta thu du.o.c . 1 cos6 x + sin6 x = (1 + 3 cos2 2x). 4D˘t t = tg2x, ta t` du.o.c a . ım . 4dx dt J= 2 2x =2 2+4 1 + 3 cos t t tg2x = arctg + C = arctg + C. 2 2
  • 53. 52 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . V´ du 5. T´ ı . ınh 3 1 J= sin 2 x cos 2 xdx. Giai. D˘t z = sin2 x ta thu du.o.c ’ a . . 1 1 J= z 1/4(1 − z)− 4 dx. 2 D´ l` t´ phˆn cua vi phˆn nhi th´.c v` o a ıch a ’ a . u a 1 m+1 +1 1 +p = 4 − = 1. n 1 4 Do vˆy ta thu.c hiˆn ph´p dˆi biˆn a . . e . e o e’ ´ 1 dz 1 − 1 = t4, − = 4t3 dt, z2 = z z2 (t4 + 1)2 v` do d´ a o t2 J = −2 dt. (t4 + 1)2 1 D˘t t = ta thu du.o.c a . . y y4 J=2 dy. (1 + y 4)2 Thu.c hiˆn ph´p t´ phˆn t`.ng phˆn b˘ng c´ch d˘t . e . e ıch a u ` a a ` a a . y3 1 u = y, dv = dy ⇒ du = dy, v=− (1 + y 4)2 4(1 + y 2) ta thu du.o.c . y 1 dy J=2 − 4) + 4(1 + y 4 1 + y4 y 1 =− 4) + J1 . 2(1 + y 2
  • 54. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 53 Dˆ t´ J1 ta biˆu diˆn tu. sˆ cua biˆu th´.c du.´.i dˆu t´ch phˆn ’ e ınh e’ ˜ e ´ ’ o ’ e’ u o a ı´ anhu. sau: 1 2 1= (y + 1) − (y 2 − 1) 2v` khi d´ a o 1y2 + 1 1 y2 − 1 J1 = dy − dy 2y4 + 1 2 y4 + 1 1 1 1 + 2 dy 1 − 2 dy 1 y 1 y = − 2 1 2 1 y2 + 2 y2 + 2 y y 1 1 d y+ d y+ 1 y 1 y = − 2 1 2 2 1 2 y− +2 y+ −2 y y 1 1 √ y− y+ − 2 1 y 1 y = √ arctg √ − √ ln + C. 2 2 2 4 2 1 √ yb + + 2 yCuˆi c`ng ta thu du.o.c ´ o u . 1 1 √ y− y+ − 2 y 1 1 y J=− + √ arctg √ 4 − √ ln +C 2(1 + y 4) 4 2 2 8 2 1 √ y+ + 2 ytrong d´ o 1 4 1 y= , t= −1, z = sin2 x. t z ` ˆ BAI TAP . T´ c´c t´ phˆn b˘ng c´ch su. dung c´c cˆng th´.c lu.o.ng gi´c ınh a ıch a ` a a ’ . a o u . adˆ biˆn dˆi h`m du.´.i dˆu t´ phˆn. ’ ´ ’ e e o a ´ o a ıch a
  • 55. 54 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 3 cos3 x 1. sin xdx. (DS. − cos x + ) 3 3x sin 2x sin 4x 2. cos4 xdx. (DS. + + ) 8 4 32 2 cos5 x 3. sin5 xdx. (DS. cos3 x − − cos x) 3 5 3 sin5 x sin7 x 4. cos7 xdx. (DS. sin x − sin3 x + − ) 5 7 x sin 4x 5. cos2 x sin2 xdx. (DS. − ) 8 32 cos5 x cos3 x 6. sin3 x cos2 xdx. (DS. − ) 5 3 sin6 x sin8 x 7. cos3 x sin5 xdx. (DS. − ) 6 8 dx 1 8. . (DS. ln |tgx|) sin 2x 2 dx π x 9. x. (DS. 3 ln tg + ) cos 4 6 3 sin x + cos x 1 x π x 10. dx. (DS. ln tg + ln tg + ) sin 2x 2 2 4 2 sin2 x tg5 x tg3 x 11. dx. (DS. + ) cos6 x 5 3 ’ ˜ Chı dˆ n. D˘t t = tgx. a a . 1 12. sin 3x cos xdx. (DS. − (cos 4x + 2 cos 2x)) 8 x 2x 3 x 1 13. sin cos dx. (DS. cos − cos x) 3 3 2 3 2 cos3 x 1 14. dx. (DS. − − sin x) sin2 x sin x sin3 x 1 15. dx. (DS. + cos x) cos2 x cos x
  • 56. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp a o a ’ ıch o a a ´ a 55 cos3 x cotg4x16. dx. (DS. − ) sin5 x 4 sin5 x 1 cos2 x17. dx. (DS. + 2 ln | cos x| − ) cos3 x 2 cos2 x 2 tg4x tg2 x18. tg5 xdx. (DS. − − ln | cos x|) 4 2 a a a . e o e’ ´ Trong c´c b`i to´n sau dˆy h˜y ´p dung ph´p dˆi biˆn a a a x 2t 1 − t2 2dt t = tg , sin x = 2 , cos x = 2 , x = 2arctgt, dx = 2 1+t 1+t 1 + t2 x dx 1 2 + tg19. . (DS. ln 2 3 + 5 cos x 4 x ) 2 − tg 2 dx 1 x π20. . (DS. √ ln tg + ) sin x + cos x 2 2 8 3 sin x + 2 cos x21. dx. 2 sin x + 3 cos x 1 (DS. (12x − 5 ln |2tgx + 3| − 5 ln | cos x|) 13 dx x22. . (DS. ln 1 + tg ) 1 + sin x + cos x 2 dx23. . (2 − sin x)(3 − sin x) x x 2 2tg − 1 1 3tg − 1 (DS. √ arctg √ 2 − √ arctg 2 √ ) 3 3 2 2 2 T´ c´c t´ phˆn dang ınh a ıch a . sinm x cosn xdx, m, n ∈ N. 1 8 124. sin3 x cos5 xdx. (DS. cos x − cos6 x) 8 6
  • 57. 56 Chu.o.ng 10. T´ phˆn bˆt dinh ´ ıch a a . 1 1 1 25. sin2 x cos4 xdx. (DS. x − sin 4x + sin2 2x ) 16 4 3 26. sin4 x cos6 xdx. 1 1 1 3 (DS. sin 8x − 8 sin 4x + sin5 2x + 8 x) 211 2 5 · 26 2 x sin 4x sin2 2x 27. sin4 x cos2 xdx. −(DS. − ) 16 64 48 1 2 1 28. sin4 x cos5 xdx. (DS. sin5 x − sin7 x + sin9 x) 5 7 9 1 1 29. sin6 x cos3 xdx. (DS. sin7 x − sin9 x) 7 9 T´ c´c t´ phˆn dang ınh a ıch a . sinα x cosβ xdx, α, β ∈ Q. sin3 x 3 √ 3 3 30. √ dx. (DS. cos x cos2 x + √ ) cos x 3 cos x 5 3 cos x ’ ˜ Chı dˆ n. D˘t t = cos x. a a . dx 3(1 + 4tg2 x) 31. √ 3 . (DS. − ) sin11 x cos x 8tg2x · 3 tg2 x ’ ˜ Chı dˆ n. D˘t t = tgx. a a . sin3 x √ 1 32. √ 3 dx. (DS. 3 3 cos x cos2 x − 1 ) cos 2x 7 √ 3 3 3 11 33. cos2 x sin3 xdx. (DS. − cos5/3 x + cos 3 x) 5 11 dx √ 34. √ 4 . (DS. 4 4 tgx) 3 5x sin x cos sin3 x 5 14 5 4 35. √ 5 dx. (DS. cos 5 x − cos 5 x) cos x 14 4
  • 58. Chu.o.ng 11T´ phˆn x´c dinh Riemann ıch a a . ’ ıch 11.1 H`m kha t´ Riemann v` t´ phˆn x´c a a ıch a a dinh . . . . . . . . . . . . . . . . . . . . . . . 58 . -. 11.1.1 Dinh ngh˜ . . . . . . . . . . . . . . . . . . 58 ıa - ` ’ ’ ıch 11.1.2 Diˆu kiˆn dˆ h`m kha t´ . . . . . . . . . . 59 e e e a . 11.1.3 C´c t´ chˆt co. ban cua t´ phˆn x´c dinh 59 a ınh a ´ ’ ’ ıch a a . 11.2 Phu.o.ng ph´p t´ a ınh t´ phˆn x´c d inh . . . 61 ıch a a . 11.3 Mˆt sˆ u.ng dung cua t´ phˆn x´c d inh . 78 . ´ o o´ . ’ ıch a a . e ıch ınh a . ’ ’ a e ıch a . ’ 11.3.1 Diˆn t´ h` ph˘ng v` thˆ t´ vˆt thˆ . . 78 e 11.3.2 T´ dˆ d`i cung v` diˆn t´ m˘t tr`n xoay 89 ınh o a . a e ıch a o . . 11.4 T´ phˆn suy rˆng . . . . . . . . . . . . . . 98 ıch a o . 11.4.1 T´ phˆn suy rˆng cˆn vˆ han . . . . . . . 98 ıch a o . a o . . ıch a o . ’ a 11.4.2 T´ phˆn suy rˆng cua h`m khˆng bi ch˘n 107 o . a .
  • 59. 58 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 11.1 a ’ ıch H`m kha t´ Riemann v` t´ phˆn a ıch a x´c dinh a . 11.1.1 -. Dinh ngh˜ ıa Gia su. h`m f (x) x´c dinh v` bi ch˘n trˆn doa n [a, b]. Tˆp ho.p h˜.u ’ ’ a a . a . a . e . a . . u n . e’ han diˆm xk k=0 : a = x0 < x1 < x2 < · · · < xn−1 < xn = b du.o.c goi l` ph´p phˆn hoach doan [a, b] v` du.o.c k´ hiˆu l` T [a, b] hay . . a e a . . a . y e a . do.n gian l` T . ’ a Dinh ngh˜ 11.1.1. Gia su. [a, b] ⊂ R, T [a, b] = {a = x0 < x1 < -. ıa ’ ’ a e a . . e ˜ · · · < xn = b} l` ph´p phˆn hoach doan [a, b]. Trˆn mˆ i doan [xj−1 , xj ], o . . . u ´ e ’m ξj v` lˆp tˆng j = 1, . . . , n ta chon mˆt c´ch t`y y diˆ o a a a o . ’ n S(f, T, ξ) = f (ξj )∆xj , ∆xj = xj − xj−1 j=1 goi l` tˆng t´ phˆn (Riemann) cua h`m f (x) theo doan [a, b] tu.o.ng . a o ’ ıch a ’ a . u ´.ng v´.i ph´p phˆn hoach T v` c´ch chon diˆm ξj , j = 1, n. Nˆu gi´.i o e a a a ’ e ´ e o . . han . n lim S(f, T, ξ) = lim f (ξj )∆xj (11.1) d(T )→0 d(T )→0 j=1 tˆn tai h˜.u han khˆng phu thuˆc v`o ph´p phˆn hoach T v` c´ch ` o . u . o . o a . e a . a a chon c´c diˆm ξj , j = 1, n th` gi´.i han d´ du.o.c goi l` t´ch phˆn x´c . a ’ e ı o . o . . a ı a a ’ a dinh cua h`m f(x). . Tˆp ho.p moi h`m kha t´ch Riemann trˆn doa n [a, b] du.o.c k´ hiˆu a . . . a ’ ı e . . y e . l` R[a, b]. a
  • 60. ’ ıch11.1. H`m kha t´ Riemann v` t´ch phˆn x´c dinh a a ı a a . 5911.1.2 - ` ’ ’ ıch Diˆu kiˆn dˆ h`m kha t´ e e . e a-. ´Dinh l´ 11.1.1. Nˆu h`m f(x) liˆn tuc trˆn doan [a, b] th` f ∈ R[a, b]. y e a e . e . ıHˆ qua. Moi h`m so. cˆp dˆu kha t´ch trˆn doan bˆt k` n˘m tron e. ’ . a a ` ´ e ’ ı e . a y ` ´ a . .p x´c dinh cua n´.trong tˆp ho a . a . ’ o .Dinh l´ 11.1.2. Gia su. f : [a, b] → R l` h`m bi ch˘n v` E ⊂ [a, b]-. y ’ ’ a a . a a . a a . .p c´c diˆm gi´n doan cua n´. H`m f(x) kha t´ch Riemannl` tˆp ho a e ’ a ’ o a ’ ı . . a ’trˆn doan [a, b] khi v` chı khi tˆp ho e a . .p E c´ dˆ do - khˆng, t´.c l` E o o o u a . . .thoa m˜n diˆu kiˆn: ∀ ε > 0, tˆn tai hˆ dˆm du.o.c (hay h˜.u han) c´c ’ a ` e e . ` . e e o . ´ . u . a ’khoang (ai , bi ) sao cho ∞ ∞ N E⊂ (ai , bi ), (bi − ai ) = lim (bi − ai ) < ε. N →∞ i=1 i=1 i=1 ´ e a ` e e . ’ . y . a e a’ ’ ı Nˆu c´c diˆu kiˆn cua dinh l´ 11.1.2 (goi l` tiˆu chuˆn kha t´ch bLo.be (Lebesgue)) du.o.c thoa m˜n th` gi´ tri cua t´ phˆn . ’ a ı a . ’ ıch a f (x)dx a o . o a . a . ’ a . a e’khˆng phu thuˆc v`o gi´ tri cua h`m f(x) tai c´c diˆm gi´n doan v` a . a ’ .o.c bˆ sung mˆt c´ch t`y y nhu.ng phaitai c´c diˆm d´ h`m f (x) du . ’ ’ . a e o a o o a . u ´ ’ a ınh . a ’ abao to`n t´ bi ch˘n cua h`m trˆn [a, b]. . e11.1.3 a ınh chˆt co. ban cua t´ phˆn x´c C´c t´ ´ a ’ ’ ıch a a dinh . a1) f (x)dx = 0. a b a 2) f(x)dx = − f(x)dx. a b ´ 3) Nˆu f, g ∈ R[a, b] v` α, β ∈ R th` αf + βg ∈ R[a, b]. e a ı
  • 61. 60 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . ´ 4) Nˆu f ∈ R[a, b] th` |f(x)| ∈ R[a, b] v` e ı a b b f(x)dx |f (x)|dx, a < b. a a ´ 5) Nˆu f, g ∈ R[a, b] th` f (x)g(x) ∈ R[a, b]. e ı ´ 6) Nˆu fg ∈ D[a, b] v` ]c, d] ⊂ [a, b] th` f (x)g(x) ∈ R[c, d]. e a ı ´ e ı o e ’ 7) Nˆu f ∈ R[a, c], f ∈ R[c, b] th` f ∈ R[a, b], trong d´ diˆm c c´ o thˆ s˘p xˆp t`y y so v´.i c´c diˆm a v` b. ’ a ´ e ´ e u ´ o a e’ a ´ Trong c´c t´ chˆt sau dˆy ta luˆn luˆn xem a < b. a ınh a a o o b ´ 8) Nˆu f ∈ R[a, b] v` f e a 0 th` ı f (x)dx 0. a ´ 9) Nˆu f, g ∈ R[a, b] v` f (x) e a g(x) ∀ x ∈ [a, b] th` ı b b f (x)dx g(x)dx. a a ´ 10) Nˆu f ∈ C[a, b], f(x) e 0, f (x) ≡ 0 trˆn [a, b] th` ∃ K > 0 sao e ı cho b f (x)dx K. a ´ 11) Nˆu f, g ∈ R[a, b], g(x) e 0 trˆn [a, b]. e M = sup f (x), m = inf f (x) [a,b] [a,b] th` ı b b b m g(x)dx f (x)g(x)dx ≤ M g(x)dx. a a a
  • 62. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 6111.2 Phu.o.ng ph´p t´ a ınh t´ ıch phˆn x´c a a dinh .Gia su. h`m f(x) kha t´ trˆn doan [a, b]. H`m ’ ’ a ’ ıch e . a x F (x) = f(x)dt, a x b adu.o.c goi l` t´ phˆn v´.i cˆn trˆn biˆn thiˆn. . . a ıch a o a . e ´ e e-.Dinh l´ 11.2.1. H`m f(x) liˆn tuc trˆn doan [a, b] l` c´ nguyˆn h`m y a e . e . a o e a e . o o . a e a ’ atrˆn doan d´. Mˆt trong c´c nguyˆn h`m cua h`m f (x) l` h`m a a x F (x) = f (t)dt. (11.2) a T´ phˆn v´.i cˆn trˆn biˆn thiˆn du.o.c x´c dinh dˆi v´.i moi h`m ıch a o a . e ´ e e . a . ´ o o . a ’ ıch e ’f (x) kha t´ trˆn [a, b]. Tuy nhiˆn, dˆ h`m F (x) dang (11.2) l` nguyˆn e e a . a e ’ e ´ ´ ` o e a ’ e .h`m cua f (x) diˆu cˆt yˆu l` f (x) phai liˆn tuc. a Sau dˆy l` dinh ngh˜ mo. rˆng vˆ nguyˆn h`m. a a . ıa ’ o . ` e e aDinh ngh˜ 11.2.1. H`m F (x) du.o.c goi l` nguyˆn h`m cua h`m-. ıa a . . a e a ’ a e . ´f (x) trˆn doan [a, b] nˆu e 1) F (x) liˆn tuc trˆn [a, b]. e . e ’ e . ’ 2) F (x) = f (x) tai c´c diˆm liˆn tuc cua f (x). . a e Nhˆn x´t. H`m liˆn tuc trˆn doan [a, b] l` tru.`.ng ho.p riˆng cua a e . a e . e . a o . e ’h`m liˆn tuc t` a e . u .ng doan. Do d´ dˆi v´.i h`m liˆn tuc dinh ngh˜a 11.2.1 ´ o o o a e . . ı .vˆ nguyˆn h`m l` tr`ng v´.i dinh ngh˜a c˜ tru.´.c dˆy v` F (x) = f (x) ` e e a a u o . ı u o a ı a ınh e . ’∀ x ∈ [a, b] v` t´ liˆn tuc cua F (x) du . .o.c suy ra t`. t´nh kha vi. u ı ’D.nh l´ 11.2.2. H`m f(x) liˆn tuc t`.ng doan trˆn [a, b] l` c´ nguyˆn-i y a e . u . e a o eh`m trˆn [a, b] theo ngh˜ cua dinh ngh˜ mo. rˆng. Mˆt trong c´c a e ıa ’ . ıa ’ o . o . a
  • 63. 62 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . nguyˆn h`m l` e a a x F (x) = f (t)dt. a Dinh l´ 11.2.3. (Newton-Leibniz) Dˆi v´.i h`m liˆn tuc t`.ng doan -. y ´ o o a e . u . trˆn [a, b] ta c´ cˆng th´ e o o u.c Newton-Leibniz: b f (x)dx = F (b) − F (a) (11.3) a trong d´ F (x) l` nguyˆn h`m cua f (x) trˆn [a, b] v´.i ngh˜ mo. rˆng. o a e a ’ e o ıa ’ o . D.nh l´ 11.2.4 (Phu.o.ng ph´p dˆi biˆn) Gia su.: -i y a o e’ ´ ’ ’ (i) f(x) x´c dinh v` liˆn tuc trˆn [a, b], a . a e . e (ii) x = g(t) x´c dinh v` liˆn tuc c`ng v´.i dao h`m cua n´ trˆn a . a e . u o . a ’ o e doan [α, β], trong d´ g(α) = a, g(β) = b v` a g(t) b. . o a Khi d´ o b β f(x)dx = f (g(t))g (t)dt. (11.4) a α D.nh l´ 11.2.5 (Phu.o.ng ph´p t´ch phˆn t`.ng phˆn). Nˆu f (x) v` -i y a ı a u ` a ´ e a g(x) c´ dao h`m liˆn tuc trˆn [a, b] th` o . a e . e ı b b b f(x)g (x)dx = f (x)g(x) − a f (x)g(x)dx. (11.5) a a CAC V´ DU ´ I . V´ du 1. Ch´.ng to r˘ng trˆn doan [−1, 1] h`m ı . u ’ a ` e . a  1  v´.i x > 0, o   f (x) = signx = 0 v´.i x = 0, x ∈ [−1, 1] o    −1 v´.i x < 0 o
  • 64. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 63 a) kha t´ b) khˆng c´ nguyˆn h`m, c) c´ nguyˆn h`m mo. rˆng. ’ ıch, o o e a o e a ’ o. ’ ’ ıch ı o a a Giai. a) H`m f (x) kha t´ v` n´ l` h`m liˆn tuc t` a e . u .ng doan. . b) Ta ch´u.ng minh h`m f(x) khˆng c´ nguyˆn h`m theo ngh˜ c˜. a o o e a ıa uThˆt vˆy moi h`m dang a a . . . a .  −x + C khi x < 0 1 F (x) = x + C2 khi x 0 ` o . a e ` a o a a a o u´dˆu c´ dao h`m b˘ng signx ∀ x = 0, trong d´ C1 v` C2 l` c´c sˆ t`y a . ı a ´ o ´ a ´y. Tuy nhiˆn, thˆm ch´ h`m “tˆt nhˆt” trong sˆ c´c h`m n`y´ e o a a a F (x) = |x| + C ´ o o . a . e’(nˆu C1 = C2 = C) c˜ng khˆng c´ dao h`m tai diˆm x = 0. Do d´ e u oh`m signx (v` do d´ moi h`m liˆn tuc t` a a o . a e . u .ng doa n) khˆng c´ dao h`m o o . a . ’ ´trˆn khoang bˆt k` ch´ e a y u .a diˆm gi´n doan. e’ a . c) Trˆn doan [−1, 1] h`m signx c´ nguyˆn h`m mo. rˆng l` h`m e . a o e a ’ o . a aF (x) = |x| v` n´ liˆn tuc trˆn doan [−1, 1] v` F (x) = f(x) khi x = 0. ı o e . e . a a √V´ du 2. T´ ı . ınh a2 − x2 dx, a > 0. 0 π ’ . ´ . ´ Giai. D˘t x = a sin t. Nˆu t chay hˆt doan 0, a e e . ´ th` x chay hˆt ı . e 2doan [0, a]. Do d´ . o a π/2 π/2 √ 1 + cos 2t a2 − x2 dx = a2 cos2 tdt = a2 dt 2 0 0 0 π/2 π/2 a2 a2 a2 π = dt + cos 2tdt = · 2 2 4 0 0V´ du 3. T´ t´ phˆn ı . ınh ıch a √ 2/2 1+x I= dx. 1−x 0
  • 65. 64 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . Giai. Ta thu.c hiˆn ph´p dˆi biˆn x = cos t. Ph´p dˆi biˆn n`y ’ . e . e o e ’ ´ e o e’ ´ a ’ a a ` thoa m˜n c´c diˆu kiˆn sau: e e. (1) x = ϕ(t) = cos t liˆn tuc ∀ t ∈ R e . √ π π 2 ´ (2) Khi t biˆn thiˆn trˆn doan e e e . , ´ th` x chay hˆt doan 0, ı . e . . √ 4 4 2 π 2 π (3) ϕ = , ϕ = 0. 4 2 2 π π (4) ϕ (t) = − sin t liˆn tuc ∀ t ∈ e . , . 4 2 Nhu. vˆy ph´p dˆi biˆn thoa m˜n dinh l´ 11.2.4 v` do d´ a . e o e ’ ´ ’ a . y a o x = cos t, dx = − sin tdt, √ π π 2 ϕ = 0, ϕ = · 2 4 2 Nhu. vˆy a . π π 4 2 t I= cotg (− sin t)dt = (1 + cos t)dt 2 −π 2 π 4 √ π/2 π 2 = t + sin t π/4 = +1− ·. 4 2 V´ du 4. T´ t´ phˆn ı . ınh ıch a √ 3/2 dx I= √ · x 1 − x2 1/2 Giai. Ta thu.c hiˆn ph´p dˆi biˆn ’ . e . e o e’ ´ x = sin t ⇒ dx = cos tdt v` biˆu th´.c du.´.i dˆu t´ phˆn c´ dang a e ’ u ´ o a ıch a o .   dt  ´ nˆu cos t > 0, e cos tdt sin t √ = sin t cos2 t − dt  ´ nˆu cos t < 0. e sin t
  • 66. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 65C´c cˆn α v` β cua t´ phˆn theo t du.o.c x´c dinh bo.i a a . a ’ ıch a . a . ’ 1 π = sin t ⇒ α = , √2 6 3 π = sin t ⇒ β = · 2 3 5π 2π ’ ´(Ta c˜ng c´ thˆ lˆy α1 = u o e a v` β1 = a ). Trong ca hai tru.`.ng ho.p ’ o . 6 3 √ 1 3 e´ ` e . e ´biˆn x = sin t dˆu chay hˆt doan [a, b] = , . ´ ´ . Ta s˜ thˆy kˆt qua e a e ’ 2 2t´ phˆn l` nhu. nhau. Thˆt vˆy trong tru.`.ng ho.p th´. nhˆt ta c´ ıch a a a a . . o . u a´ ocos t > 0 v` a π/3 π/3 √ dt t 2+ 3 I= = ln tg = ln · sin t 2 3 π/6 π/6 5π 2π Trong tru.`.ng ho.p th´. hai t ∈ o . u , ta c´ cos t < 0 v` o a 6 3 2π/3 √ dt t 2π/3 2+ 3 I =− = − ln tg = ln · sin t 2 5π/6 3 5π/6V´ du 5. T´ t´ phˆn ı . ınh ıch a π/3 x sin x I= dx. cos2 x 0 Giai. Ta t´ b˘ng phu.o.ng ph´p t´ phˆn t`.ng phˆn. ’ ` ınh a a ıch a u ` a D˘t a . u = x ⇒ du = dx, sin xdx 1 dv = 2x ⇒v= · cos cos x
  • 67. 66 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . Do d´ o π/3 1 dx π/3 π x π π/3 I =x· −= π − ln tg 2 + 4 cos x 0 cos x 3 cos 0 0 3 2π π π π 2π 5π = − ln tg + + ln tg = − ln tg · 3 6 4 4 3 12 V´ du 6. T´ t´ phˆn ı . ınh ıch a 1 I= x2(1 − x)3 dx. 0 ’ Giai. Ta d˘t a . u = x2 , dv = (1 − x)3dx ⇒ (1 − x)4 du = 2xdx, v=− · 4 Do d´ o 1 2 (1 − x)4 1 (1 − x)4 I = −x + 2x dx = 0 + I1. 4 0 4 0 I1 T´ I1. T´ch phˆn t`.ng phˆn I1 ta c´ ınh ı a u ` a o 1 1 1 4 1 (1 − x)5 1 1 (1 − x)5 I1 = x(1 − x) dx = − x + dx 2 2 5 0 2 5 0 0 1 (1 − x)6 1 1 1 =0− = ⇒I= · 10 6 0 60 60 V´ du 7. Ap dung cˆng th´.c Newton-Leibnitz dˆ t´nh t´ch phˆn ı . ´ . o u ’ e ı ı a 100π 1 √ 1) I1 = 1 − cos 2xdx, 2) I2 = ex arc sin(e−x )dx. 0 0
  • 68. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 67 √ √ ’ Giai. Ta c´ o 1 − cos 2x = 2| sin x|. Do d´ o 100π 100π √ √ 1 − cos 2xdx = 2 | sin x|dx 0 0 π 2π 3π √ = 2 sin xdx − sin xdx + sin xdx − . . . 0 π 2π 100π + ··· + sin xdx 99π √ √ = − 2[2 + 2 + · · · + 2] = 200 2. 2) Thu.c hiˆn ph´p dˆi biˆn t = e−x , sau d´ ´p dung phu.o.ng ph´p . e. e o e ’ ´ oa . at´ phˆn t` ıch a u .ng phˆn. Ta c´ ` a o arc sin t exarc sin(e−x )dx = − dt t2 1 dt = arc sin t − √ t t 1 − t2 1 = arc sin t + I1 . t 1 dt d 1 1 I1 = − √ = t = ln + − 1 + C. t 1 − t2 1 2 t t2 −1 tDo d´ o arc sin t 1 1 ex arc sin e−x dx = + ln + −1 +C t t t2 √ = exarc sin e−x + ln(ex + e2x − 1) + CNguyˆn h`m v`.a thu du.o.c c´ gi´.i han h˜.u han tai diˆm x = 0. do e a u . o o . u . . ’ ed´ theo cˆng th´.c (11.3) ta c´ o o u o 1 π √ ex arc sin e−x dx = earc sin e−1 − + ln(e + e2 − 1). 2 0
  • 69. 68 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . V´ du 8. T´ t´ phˆn Dirichlet ı . ınh ıch a π/2 sin(2n − 1)x dx, n ∈ N. sin x 0 Giai. Ta c´ cˆng th´.c ’ o o u n−1 1 sin(2n − 1)x + cos 2kx = · 2 k=1 2 sin x π/2 T`. d´ v` lu.u y r˘ng u o a ´ a` cos 2kxdx = 0, k = 1, 2, . . . , n − 1 ta c´ o 0 π/2 sin(2n − 1)x π dx = · sin x 2 0 ` ˆ BAI TAP . T´ c´c t´ phˆn sau dˆy b˘ng phu.o.ng ph´p dˆi biˆn (1-14). ınh a ıch a a a ` a o e’ ´ 5 xdx 1. √ . (DS. 4) 1 + 3x 0 ln 3 3 dx ln 2. . (DS. 2) ex − e−x 2 ln 2 √ 3 (x3 + 1)dx 7 3. √ . (DS. √ − 1). D˘t x = 2 sin t. a . x2 4 − x2 2 3 1 π/2 dx π 4. . (DS. √ ) 2 + cos x 3 3 0
  • 70. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 69 1 x2 dx 15. . (DS. ) (x + 1)4 24 0 ln 2 √ 4−π6. ex − 1dx. (DS. ) 2 0 √ 7 x3dx7. . (DS. 3) √ 3 (x2 + 1)2 3 ’ ˜ Chı dˆ n. D˘t t = x2 + 1. a a . e√ 4 1 + ln x √8. dx. (DS. 0, 8(2 4 2 − 1)) x 1 ’ a˜ Chı dˆ n. D˘t t = 1 + ln x. a . √ + 3 √ 81π9. x2 9 − x2dx. (DS. ) 8 −3 ’ a˜ chı dˆ n. D˘t x = 3 cos t. a . 3 x 3(π − 2)10. dx. (DS. ) 6−x 2 0 Chı dˆ n. D˘t x = 6 sin2 t. ’ a˜ a . π x 5π11. sin6 dx. (DS. ) 2 16 0 ’ a˜ Chı dˆ n. D˘t x = 2t. a . π/4 812. cos7 2xdx. (DS. ) 35 0 t ’ a˜ Chı dˆ n. D˘t x = a . 2
  • 71. 70 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . √ 2/2 √ 1+x π 2 13. dx. (DS. + 1 − ) 1−x 4 2 0 ’ a˜ Chı dˆ n. D˘t x = cos t. a . 29 √ 3 (x − 2)2 3 3 14. dx. (DS. 8 + π) 3+ 3 (x − 2)2 2 3 T´ c´c t´ phˆn sau dˆy b˘ng phu.o.ng ph´p t´ch phˆn t`.ng ınh a ıch a a ` a a ı a u ` phˆn (15-32). a 1 1 15. x3 arctgxdx. (DS. ) 6 0 e 16. | ln x|dx. (DS. 2(1 − 1/e)) 1/e π 1 π 17. ex sin xdx. (DS. (e + 1)) 2 0 1 e2 + 3 18. x3 e2xdx. (DS. ) 8 0 1 arc sin x √ 19. √ dx. (DS. π 2 − 4) 1+x 0 π/4 π ln 2 20. ln(1 + tgx)dx. (DS. ) 8 0 π/b b πa 21. eax sin bxdx. (DS. e b +1 ) a2 + b2 0 1 1+e 22. e−x ln(ex + 1)dx. (DS. − ln(e + 1) + 2 ln 2 + 1) e 0
  • 72. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 71 π/2 π23. sin 2x · arctg(sin x)dx. (DS. − 1) 2 0 2 124. sin(ln x)dx. (DS. sin(ln 2) − cos(ln 2) + ) 2 1 π25. x3 sin xdx. (DS. π 3 − 6π) 0 2 326. xlog2xdx. (DS. 2 − ) 4 ln 2 1 √ a 7 √ x3 141a3 3 a27. √ 3 dx. (DS. ) a2 + x2 20 0 a √ πa228. a2 − x2 dx. (DS. ) 4 0 π/2 x + sin x π √29. dx. (DS. (1 + 3)) 1 + cos x 6 π/6 π/2 mπ cos30. sinm x cos(m + 2)xdx. (DS. − 2 ) m+1 0 π/231. cosm x cos(m + 2)xdx. (DS. 0) 0 π/2 π32. cos x cos 2nxdx. (DS. (−1)n−1 ) 4n 0
  • 73. 72 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 2 33. T´ ınh f(x)dx, trong d´ o 0   x2 khi 0 x 1 f(x) = 2 − x khi 1 x 2 b˘ng hai phu.o.ng ph´p; a) su. dung nguyˆn h`m cua f (x) trˆn doa n ` a a ’ . e a ’ e . 5 [0, 2]; b) chia doan [0, 2] th`nh hai doan [0, 1] v` [1, 2]. (DS. ) . a . a 6 34. Ch´ u.ng minh r˘ng nˆu f (x) liˆn tuc trˆn doan [− , ] th` ` a ´ e e . e ı . (i) f(x)dx = 2 ˜ f(x)dx khi f (x) l` h`m ch˘ n; a a a − 0 (ii) ’ f(x)dx = 0 khi f (x) l` h`m le. a a − 35. Ch´.ng minh r˘ng ∀ m, n ∈ Z c´c d˘ng th´.c sau dˆy du.o.c thoa u ` a a a ’ u a . ’ m˜n: a π (i) sin mx cos nxdx = 0. −π π (ii) cos mx cos nxdx = 0, m = n. −π π (iii) sin mx sin nxdx = 0, m = n. −π 36. Ch´.ng minh d˘ng th´.c u ’ a u b b f(x)dx = f (a + b − x)dx. a a ’ ˜ Chı dˆ n. D˘t x = a + b − t. a a .
  • 74. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 7337. Ch´.ng minh d˘ng th´.c u ’ a u π/2 π/2 f(cos x)dx = f (sin x)dx. 0 0 π ’ a˜ Chı dˆ n. D˘t t = − x. a . 238. Ch´ u.ng minh r˘ng nˆu f(x) liˆn tuc khi x ` a ´ e e . 0 th` ı a a2 1 x3 f(x2 )dx = xf(x)dx. 2 0 0 x39. Ch´.ng minh r˘ng nˆu f(t) l` h`m le th` u ` a ´ e a a ’ ı ˜ f (t)dt l` h`m ch˘ n, a a a at´.c l` u a −x x f(t)dt = f (t)dt. a a ’ ˜a a . a e ’ ˜ Chı dˆ n. D˘t t = −x v` biˆu diˆn e −x a −x f(t)dt = + −a −a av` su. dung t´ ch˘ n le cua h`m f. a ’ . ınh ˜ ’ ’ a a T´ c´c t´ phˆn sau dˆy (40-65) b˘ng c´ch ´p dung cˆng th´.c ınh a ıch a a ` a a a . o uNewton-Leibnitz. 5 xdx40. √ . (DS. 4) 1 + 3x 0 ln 3 dx ln 1, 541. . (DS. ) ex − e−x 2 ln 2
  • 75. 74 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . √ 3 (x3 + 1)dx 7 42. √ . (DS. √ − 1) x2 4 − x2 2 3 0 π/2 dx π 43. . (DS. √ ) 2 + cos x 3 3 0 ln 2 √ 4−π 44. ex − 1dx. (DS. ) 2 0 √ 7 x3dx 45. . (DS. 3) √ 3 (x2 + 1)2 3 e √ 4 1 + ln x √ 46. dx. (DS. 0, 8(2 4 2 − 1)) x 1 3 √ 81π 47. x2 9 − x2dx. (DS. ) 8 −3 3 x 3(π − 2) 48. dx. (DS. ) 6−x 2 0 Chı dˆ n. D˘t x = 6 sin2 t. ’ a˜ a . 4 x2 + 3 11 49. dx. (DS. + 7ln2) x−2 2 3 −1 x+1 4 1 50. dx. (DS. 2 ln − ) x2 (x − 1) 3 2 −2 1 (x2 + 3x)dx π 51. . (DS. ) (x + 1)(x2 + 1) 4 0
  • 76. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 75 1 √ dx 2+ 552. √ . (DS. ln √ ) x2 + 2x + 2 1+ 2 0 4 dx53. √ . (DS. 2 − ln 2) 1 + 2x + 1 0 2 1 ex 1 154. dx. (DS. (e − e 4 )) x3 2 1 e dx π55. . (DS. ) x(1 + ln2 x) 4 1 e cos(ln x)56. dx. (DS. sin 1) x 1 1 257. xe−xdx. (DS. 1 − ) e 0 π/3 √ xdx π(9 − 4 3)58. . (DS. ) sin2 x 36 π/4 359. ln xdx. (DS. 3 ln 3 − 2) 1 2 360. x ln xdx. (DS. 2 ln 2 − ) 4 1 1/2 √ π 361. arc sin xdx. (DS. + − 1) 12 2 0 π62. x3 sin xdx. (DS. π 3 − 6π) 0
  • 77. 76 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . π/2 eπ − 2 63. e2x cos xdx. (DS. ) 5 0 2 64. |1 − x|dx. (DS. 1) 0 b |x| 65. dx. (DS. |b| − |a|) x a T´ c´c t´ phˆn sau dˆy ınh a ıch a a a/b dx π 66. = a2 +b 2x 4ab 0 1 x2 dx 9√ 64 67. √ = 6− 4 + 2x 5 15 0 2 dx 1 5 68. = ln x2 + 5x + 4 3 4 0 1 dx 2π 69. = √ x2 −x+1 3 3 0 1 (x2 + 1) π 70. 4 + x2 + 1 dx = √ x 2 2 0 pi/2 dx 71. =1 1 + cos x 0 1 √ 1 1 √ 72. x2 + 1dx = √ + ln(1 + 2) 2 2 0
  • 78. 11.2. Phu.o.ng ph´p t´ t´ phˆn x´c d .nh a ınh ıch a a i 77 1 √ 3 3/2 3π73. 1− x2 dx = 32 0 D˘t x = sin3 ϕ. a . a a−x π 2 274. x2 dx = − a , a > 0. a+x 4 3 0 D˘t x = a cos ϕ. a . 2a √ πa275. 2ax − x2 dx = 2 0 D˘t x = 2a sin2 ϕ. a . 1 ln(1 + x) π76. 2 dx = ln 2. 1+x 8 0 Chı dˆ n. D˘t x = tgt rˆi ´p dung cˆng th´.c ’ a˜ a . ` a o . o u √ π sin t + cos t = 2 cos −t 4 π x sin x π277. dx = 1 + cos2 x 4 0 π π/2 π ’ a˜ ’ ˜ Chı dˆ n. Biˆu diˆn e e = + rˆi thu.c hiˆn ph´p dˆi biˆn trong ` o . e . e o e’ ´ 0 0 π/2t´ phˆn t`. π/2 dˆn π. ıch a u ´ e π √ 378. sin xdx = 0 −π π 279. ex sin xdx = 0 −π
  • 79. 78 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . π/2 π 80. (cos2 x + x2 sin x)dx = 2 −π/2 1 81. (ex + e−x )tgxdx = 0 −1 pi/2 1 82. sin x sin 2x sin 3xdx = 6 0 e 83. | ln x|dx = 2(1 − e−1 ) 1/e π 3 84. ex cos2 xdx = (eπ − 1) 5 0 e 85. (1 + ln x)2dx = 2e − 1 1 Chı dˆ n. T´ phˆn t`.ng phˆn. ’ ˜a ıch a u ` a 11.3 Mˆt sˆ u.ng dung cua t´ phˆn x´c . ´ o o´ . ’ ıch a a dinh . 11.3.1 e ıch ınh . ’ a ’ a e ıch a . ’ Diˆn t´ h` ph˘ng v` thˆ t´ vˆt thˆ e e ıch ı . ’ 1 Diˆn t´ h`nh ph˘ng a 1+ . Diˆn t´ch h` thang cong D gi´.i han bo.i du.`.ng cong L c´ e ı . ınh o . ’ o o phu.o.ng tr` y = f (x), f(x) ınh 0 ∀ x ∈ [a, b] v` c´c du.`.ng th˘ng a a o ’ a
  • 80. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 79x = a, x = b v` truc Ox du.o.c t´ theo cˆng th´.c a . . ınh o u b SD = f (x)dx. (11.6) a ´ Nˆu f(x) e 0 ∀ x ∈ [a, b] th` ı b SD = − f (x)dx (11.6*) a ´ ` Nˆu d´y h` thang cong n˘m trˆn truc Oy th` e a ınh a e . ı d SD = g(y)dy, x = g(y), y ∈ [c, d]. c 2+ Nˆu du.`.ng cong L du.o.c cho bo.i phu.o.ng tr`nh tham sˆ x = ϕ(t), ´ e o . ’ ı ´ oy = ψ(t), t ∈ [α, β] th` ı β SD = ψ(t)ϕ (t)dt. (11.7) α 3+ Diˆn t´ cua h` quat gi´.i han bo.i du.`.ng cong cho du.´.i dang e ıch ’ ınh . . o . ’ o o .toa dˆ cu.c ρ = f(ϕ) v` c´c tia ϕ = ϕ0 v` ϕ = ϕ1 du.o.c t´nh theo cˆng . o .. a a a . ı oth´.c u ϕ1 1 SQ = [f (ϕ)]2dϕ. (11.8) 2 ϕ0 4+ Nˆu miˆn D = {(x, y) : a ´ e ` e x b; f1(x) y f2 (x)} th` ı b SD = [f2(x) − f1(x)]dx. (11.9) a
  • 81. 80 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . ’ 2. Thˆ t´ch vˆt thˆ e ı a . e’ 1+ Nˆu biˆt du.o.c diˆn t´ch S(x) cua thiˆt diˆn tao nˆn bo.i vˆt thˆ ´ e ´ e . e ı . ’ ´ . e e . e ’ a . ’ e v` m˘t ph˘ng vuˆng g´c v´.i truc Ox tai diˆm c´ ho`nh dˆ x th` khi a a . ’ a o o o . . e’ o a o . ı ’ . .o.ng b˘ng dx th` vi phˆn cua thˆ t´ch b˘ng x thay dˆi mˆt dai lu . o o . ` a ı a ’ ’ e ı ` a dv = S(x)dx, v` thˆ t´ to`n vˆt thˆ du.o.c t´nh theo cˆng th´.c ’ a e ıch a a . ’ . ı e o u b V = S(x)dx (11.10) a a ınh ´ e o o ’ a . ’ trong d´ [a, b] l` h` chiˆu vuˆng g´c cua vˆt thˆ lˆn truc Ox. o e e . + ´ . 2 Nˆu vˆt thˆ e a ’ du.o.c tao nˆn do ph´p quay h`nh thang cong gi´.i e . . e e ı o han bo ’.i du.`.ng cong y = f (x), f (x) o 0 ∀x ∈ [a, b], truc Ox v` c´c a a . . du.`.ng th˘ng x = a, x = b xung quanh truc Ox th` diˆn t´ vˆt thˆ o ’ a . ı e ıch a . . e’ tr`n xoay d´ du.o.c t´ theo cˆng th´.c o o . ınh o u b Vx = π [f (x)]2dx. (11.11) a ´ Nˆu quay h` thang cong xung quanh truc Oy th` vˆt tr`n xoay e ınh . ı a o . .o.c c´ thˆ t´ch thu du . o e ı ’ d Vy = π [x(y)]2dy, x = x(y); [c, d] = prOy V. (11.12) c 3+ Nˆu h`m y = f (x) du.o.c cho bo.i c´c phu.o.ng tr`nh tham sˆ ´ e a . ’ a ı ´ o x = x(t) y = y(t), t ∈ [α, β]
  • 82. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 81thoa m˜n nh˜.ng diˆu kiˆn n`o d´ th` thˆ t´ch vˆt thˆ tao nˆn bo.i ’ a u ` e e a o ı e ı . ’ a . ’ e . e ’ph´p quay h` thang cong xung quanh truc Ox b˘ng e ınh . ` a β Vx = π y 2 (t)x (t)dt (11.13) α 4+ Nˆu h` thang cong du.o.c gi´.i han bo.i c´c du.`.ng cong 0 ´ e ınh . o . ’ a oy1(x) y2 (x) ∀ x ∈ [a, b], trong d´ y1(x) v` y2 (x) liˆn tuc trˆn [a, b] o a e . e ’ t´ vˆt thˆ tao nˆn do ph´p quay h`nh thang d´ xung quanhth` thˆ ıch a ı e . ’ . e e e ı otruc Ox b˘ng . ` a b Vx = π (y2 (x))2 − (y1(x))2 dx. (11.14) a 5+ Dˆi v´.i vˆt thˆ thu du.o.c bo.i ph´p quay h`nh thang cong xung ´ o o a . ’ e . ’ e ı o o ` .o.ng tu. ta c´ . a ’ . ´ equanh truc Oy v` thoa m˜n mˆt sˆ diˆu kiˆn tu a e . . o β Vy = π x2(t)y (t)dt (11.15) α d Vy = π (x2(y))2 − (x1 (y))2 dy. (11.16) c CAC V´ DU ´ I .V´ du 1. T` diˆn t´ h` ph˘ng gi´.i han bo.i du.`.ng astroid ı . ım e ıch ınh . ’ a o . ’ o 3x = a cos3 t, y = a sin t. Giai. Ap dung cˆng th´.c (11.7). V` du.`.ng astroid dˆi x´.ng qua ’ ´ . o u ı o ´ o u
  • 83. 82 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . c´c truc toa dˆ (h˜y v˜ h`nh !) nˆn a . . o a e ı . e 0 S = 4S1 = 4 a sin3 t · 3a cos2 t(− sin t)dt π/2 π/2 = 12a2 sin4 t cos2 tdt 0 π/2 3 = a2 (1 − cos 2t)(1 − cos2 2t)dt 2 0 3 3πa = 8 V´ du 2. Trˆn hypecbon x2 − y 2 = a2 cho diˆm M(x0 , y0 ) x0 > 0, ı . e e’ y0 > 0. T´ diˆn t´ h`nh ph˘ng gi´.i han bo.i truc Ox, hypecbˆn v` ınh e ıch ı . a’ o . ’ . o a tia OM. Giai. Ta chuyˆn sang toa dˆ cu.c theo cˆng th´.c x = r cos ϕ, ’ ’ e . o . . o u y = r sin ϕ. Khi d´ phu o .o.ng tr`nh hypecbˆn c´ dang ı o o . 2 a2 a2 r = = · cos2 ϕ − sin2 ϕ cos 2ϕ y0 D˘t tgα = a . v` lu.u y r˘ng x2 − y0 = a2 ta thu du.o.c a ´ a` 0 2 . x0 α α 1 a2 dϕ a2 1 + tgα S= r2 dϕ = = ln 2 2 cos 2ϕ 4 1 − tgα 0 0 2 a (x0 + y0)2 a2 x0 + y0 = ln = ln · 4 a2 2 a ’. a O dˆy ta d˜ su. dung cˆng th´.c a ’ . o u dt t π = ln tg + + C. cos t 2 4
  • 84. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 83V´ du 3. T´ diˆn t´ch h` ph˘ng gi´.i han bo.i c´c du.`.ng c´ phu.o.ng ı . ınh e ı . ınh a ’ o . ’ a o o 2 2 2 2tr` x + y = 2y, x + y = 4y; y = x v` y = −x. ınh a Giai. Du.a phu.o.ng tr` du.`.ng tr`n vˆ dang ch´nh t˘c ta c´: ’ ınh o o ` . e ı ´ a o 2 2 2 2x + (y − 1) = 1 v` x + (y − 2) = 4. D´ l` hai du o a o a .`.ng tr`n tiˆp x´c o ´ e utrong tai tiˆp diˆm O(0, 0). T`. d´ miˆn ph˘ng D gi´.i han bo.i c´c . e ´ e’ u o ` e a’ o . ’ adu.`.ng d˜ cho dˆi x´.ng qua truc Oy. L`.i giai s˜ du.o.c do.n gian ho.n o a ´ o u . o ’ e . ’ ´ ’nˆu ta chuyˆn sang toa dˆ cu e e .c (v´.i truc cu.c tr`ng v´.i hu.´.ng du.o.ng . o . . o . . u o o ’cua truc ho`nh): . a x = r cos ϕ x2 + y 2 = 2y ⇒ r = 2 sin ϕ, ⇒ y = r sin ϕ x2 + y 2 = 4y ⇒ r = 4 sin ϕ,v` a π 3π D = (r, ϕ) : ϕ ; 2 sin ϕ r 4 sin ϕ . 4 4K´ hiˆu S ∗ l` diˆn t´ phˆn h` tr`n gi´.i han bo.i du.`.ng tr`n x2 + y e. a e ıch ` . a ınh o o . ’ o o π 3πy 2 = 4y (t´.c l` r = 4 sin ϕ) v` hai tia ϕ = u a a v` ϕ = a ; S l` diˆn a e . 4 4t´ phˆn h` tr`n gi´ . ıch a ınh o o.i han bo.i x2 + y 2 = 2y (t´.c l` r = 2 sin ϕ) v` ’ u a ahai tia d˜ nˆu. Khi d´ a e o π/2 π/2 1 1 SD = S ∗ − S = 2 (4 sin ϕ)2 dϕ − (2 sin ϕ)2 dϕ 2 2 π/4 π/4 π/2 3π = 12 sin2 ϕdϕ = + 3. 2 π/4 ı . ’V´ du 4. T´ thˆ t´ vˆt tr`n xoay tao nˆn do ph´p quay h` ınh e ıch a . o . e e ınh 2 2 x ythang cong gi´.i han bo.i c´c du.`.ng y = ±b, 2 − 2 = 1 xung quanh o . ’ a o a btruc Oy. . Giai. Do t´ dˆi x´.ng cua vˆt tr`n xoay dˆi v´.i m˘t ph˘ng xOz ’ ´ ınh o u ’ a o . ´ o o a . ’ a(ban doc h˜y tu. v˜ h` . ’ ` ı ’. e ’ a ’ . a . e ınh) ta chı cˆn t´nh nu a bˆn phai m˘t ph˘ng xOz a . a
  • 85. 84 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . a ’ l` du. Ta c´ o b b 2 2 y2 V = 2V1 = 2π x dy = 2πa 1+ dy b2 0 0 y3 b 8 = 2πa2 y + 2 = πa2b. 3b 0 3 ı . ’ ınh e ıch a . ’ . V´ du 5. T´ thˆ t´ vˆt thˆ lˆp nˆn do quay astroid x = a cos3 t, e a e y = a sin3 t, 0 t 2π xung quanh truc Ox. . Giai. Du.`.ng astroid dˆi x´.ng dˆi v´.i c´c truc Ox v` Oy. Do d´ ’ o ´ o u ´ o o a . a o a a 2 Vx = π y dx = 2π y 2 dx −a 0 y 2 = a2 sin6 t, dx = −3a cos2 t sin tdt π t = khi x = 0, t = 0 khi x = a. 2 Do d´ o a 0 V = 2π y 2dx = −6a3π sin6 t cos2 t sin tdt 0 π/2 0 = 6a3π (1 − cos2 t)3 cos2 t(− sin tdt) π/2 0 3 = 6a π (cos2 t − 3 cos4 t + 3 cos6 t − cos8 t)(d(cos t) π/2 32 3 = ··· = πa . 105 V´ du 6. T´ thˆ t´ vˆt thˆ gi´.i han bo.i hypecboloid mˆt tˆng ı . ’ ınh e ıch a . ’ e o . ’ o ` . a x2 y 2 z 2 + 2 − 2 =1 a2 b c
  • 86. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 85 a. a’v` c´c m˘t ph˘ng z = 0, z = h (h > 0). a a Giai. Ta s˜ ´p dung cˆng th´.c (11.10), trong d´ ta x´t c´c thiˆt ’ ea . o u o e a ´ ediˆn tao nˆn bo.i c´c m˘t ph˘ng vuˆng g´c v´.i truc Oz. Khi d´ (11.10) e . e ’ a . a . ’ a o o o . oc´ dang o . h V = S(z)dz, 0 a e ıch ’ . ´ . e e . o a . ´ .trong d´ S(z) l` diˆn t´ cua thiˆt diˆn phu thuˆc v`o z. Khi c˘t vˆt o a athˆ bo.i m˘t ph˘ng z = const ta thu du.o.c elip v´.i phu.o.ng tr`nh ’ e ’ a . a’ . o ı    x2 y2 x 2 y 2 2 z    + =1 + 2 =1+ 2 z2 z2 a 2 b c ⇔ a2 1 + 2 b2 1+ z = const    c c2  z = constT`. d´ suy r˘ng u o ` a z2 z2 a1 = a2 1+ 2 , b1 = b 2 1+ c c2l` c´c b´n truc cua elip. Nhu.ng ta biˆt r˘ng diˆn t´ h` elip v´.i a a a . ’ ´ ` e a e ıch ınh . o a ’ o e ı `b´n truc a1, b1 l` πa1b1 (c´ thˆ t´nh b˘ng cˆng th´ a a o u .c (11.7) dˆi v´.i elip ´ o o .c´ phu.o.ng tr` tham sˆ x = a1 cos t, y = b1 sin t, t ∈ [0, 2π]). o ınh o´ Nhu. vˆy a . z2 S(z) = πab 1 + 2 , z ∈ [0, h]. c . d´ theo cˆng th´.c (11.10) ta c´T` o u o u o h z2 h2 V = πab 1 + dz = πabh 1 + 2 . c2 3c 0V´ du 7. T´ thˆ t´ vˆt thˆ thu du.o.c bo.i ph´p quay h`nh ph˘ng ı . ’ ınh e ıch a . ’ e . ’ e ı ’ a o.i han bo.i du.`.ng y = 4 − x2 v` y = 0 xung quanh du.`.ng th˘nggi´ . ’ o a o ’ ax = 3 (h˜y v˜ h` a e ınh).
  • 87. 86 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . Giai. Vˆt tr`n xoay thu du.o.c c´ t´nh chˆt l` moi thiˆt diˆn tao ’ a. o . o ı ´ a a . ´ e . e . ’ bo.i m˘t ph˘ng vuˆng g´c v´.i truc quay dˆu l` v`nh tr`n gi´.i han bo.i a a’ o o o ` a a e o o . ’ . . c´c du.`.ng tr`n dˆng tˆm. X´t thiˆt diˆn c´ch gˆc toa dˆ khoang b˘ng a o o ` o a e ´ . e e a ´ o . o . ’ ` a y (0 y 4). Ta c´ o S = πR2 − πr2 = π[(3 + x)2 − (3 − x)2] = 12πx = 12π 4−y a a o ’ . e’ e o a ’ v` x l` ho`nh dˆ cua diˆm trˆn parabˆn d˜ cho. Khi y thay dˆi dai ı o . lu.o.ng dy th` vi phˆn thˆ t´ch . ı a ’ e ı dv = S(y)dy = 12π 4 − ydy. ’ o e ıch a a ` Do d´ thˆ t´ to`n vˆt b˘ng . a 4 0 V = 12π 4 − ydy = 8π(4 − y)3/2 = 64π. 4 0 V´ du 8. T` thˆ t´ vˆt thˆ gi´.i han bo.i c´c m˘t x2 + y 2 = R2 ; ı . ’ ım e ıch a . ’ e o . ’ a a . x z x z y = 0, z = 0, + − 1 = 0, − − 1 = 0. R h R h ’ Giai. Do t´ dˆi x´ ınh o u´ .ng (h˜y v˜ h` a e ınh) cua vˆt thˆ dˆi v´.i m˘t ’ a ’ ´ e o o a . . ph˘’ ng x = 0 nˆn ta chı cˆn t´ thˆ t´ch phˆn n˘m trong g´c phˆn a e ’ ` ınh e ı a ’ ` ` a a o ` a t´m th´ a u. nhˆt. Moi thiˆt diˆn tao nˆn bo.i c´c m˘t ph˘ng ⊥ Ox dˆu a´ ´ e . e e ’ a a ’ a ` e . . . l` h` ch˜ a ınh u a . nhˆt ABCD v´.i OA = x. Khi d´ o o . h √ S(x) = SABCD = AB · AD = (R − x) · R2 − x2. R T`. d´ thu du.o.c u o . R R h √ V =2 S(x)dx = 2 (R − x) R2 − x2dx (d˘t x = R sin t) a . R 0 0 π/2 hR2 (3π − 4) = 2hR2 (1 − sin t) cos2 tdt = · 6 0
  • 88. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 87 ` ˆ BAI TAP . a a a a ınh e ı . a ı ’ Trong c´c b`i to´n sau dˆy (1-17) t´ diˆn t´ch c´c h`nh ph˘ng agi´.i han bo.i c´c du.`.ng d˜ chı ra. o . ’ a o a ’ 91. y = 6x − x2 − 7, y = x − 3. (DS. ) 2 22. y = 6x − x , y = 0. (DS. 36) 53. 4y = 8x − x2 , 4y = x + 6. (DS. 5 ) 24 2 24. y = 4 − x , y = x − 2x. (DS. 9)5. 6x = y 3 − 16y, 24x = y 3 − 16y. (DS. 16)6. y = 1 − ex , x = 2, y = 0. (DS. e2 − 3) 17. y = x2 − 6x + 10, y = 6x − x2 ; x = −1. (DS. 21 ) 3 π8. y = arc sin x, y = ± , x = 0. (DS. 2) 2 (e − 1)29. y = ex , y = e−x , x = 1. (DS. ) e 410. y 2 = 2px, x2 = 2py. (DS. p2 ) 311. x + y + 6x − 2y + 8 = 0, y = x2 + 6x + 10 2 2 3π + 2 9π − 2 (DS. S1 = , S2 = ) 6 612. x = a(t − sin t), y = a(1 − cos t), t ∈ [0, 2π]. (DS. 3πa2) Chı dˆ n. Dˆy l` phu.o.ng tr` tham sˆ cua du.`.ng xycloid. ˜ ’ a a a ınh ´ o ’ o 3πa213. x = a cos3 t, y = a sin3 t, t ∈ [0, 2π]. (DS. ) 814. x = a cos t, y = b sin t, t ∈ [0, 2π]. (DS. πab)15. Du.`.ng lemniscate Bernoulli ρ2 = a2 cos 2ϕ. (DS. a2 ) o16. Du.`.ng h` tim (Cacdioid) ρ = a(1 + cos ϕ). o ınh 3πa2 (DS. ) 2
  • 89. 88 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . √ 17∗ C´c du.`.ng tr`n ρ = 2 3a cos ϕ, ρ = 2a sin ϕ. a o o 5 √ (DS. a2 π − 3 ) 6 ’ e ı a . ’ Trong c´c b`i to´n sau (18-22) h˜y t´nh thˆ t´ch vˆt thˆ theo diˆn a a a a ı e e . t´ c´c thiˆ ıch a ´t diˆn song song. e e . x2 y 3 z 2 4 ’ 18. Thˆ t´ h` elipxoid e ıch ınh + 2 + 2 = 1. (DS. πabc) a2 b c 3 ’ ’ 19. Thˆ t´ vˆt thˆ gi´ . e ıch a e o.i han bo.i m˘t tru x2 + y 2 = a2 , y 2 + z 2 = a2. ’ a . . . 16 3 (DS. a) 3 Chı dˆ n. Do t´ dˆi x´.ng, chı cˆn t´ thˆ t´ch mˆt phˆn t´m ’ a ˜ ´ ınh o u ’ ` ınh e ı a ’ o . ` a a vˆt thˆ v´.i x > 0, y > 0, z > 0 l` du. C´ thˆ lˆy c´c thiˆt diˆn song a . ’ e o a ’ ’ ´ o e a a ´ e e . song v´ a o .i m˘t ph˘ng xOz. D´ l` c´c h`nh vuˆng. ’ a o a a ı o . 20. Thˆ t´ vˆt thˆ h` n´n v´.i b´n k´ d´y R v` chiˆu cao h. ’ e ıch a . ’ e ınh o o a ınh a a ` e πR2 h (DS. ) 3 Chı dˆ n. Dich chuyˆn h`nh n´n vˆ vi tr´ v´.i dınh tai gˆc toa dˆ ’ a ˜ . ’ e ı o ` . ı o ’ e ´ . o . o . a . o ´i x´.ng l` Ox. Thiˆt diˆn cˆn t` l` h` tr`n v´.i b´n k´nh v` truc dˆ u a e´ e ` ım a ınh o o a ı . a R r(x) = x (?). x 21. Thˆ t´ vˆt thˆ gi´.i han bo.i c´c m˘t n´n ’ e ıch a . ’ e o . ’ a a o . 2 x2 y 2 (z − 2) = + ’ v` m˘t ph˘ng z = 0. a a. a 3 2 √ 8π 6 (DS. ) 3 22. Thˆ t´ vˆt thˆ gi´.i han bo.i m˘t tru partabolic z = 4 − y 2 , c´c ’ e ıch a . ’ e o . ’ a . . a 16a . ’ . o a a . . ’ m˘t ph˘ng toa dˆ v` m˘t ph˘ng x = a. (DS. a a a ) 3 a a ı ’ e ı ’ a o Trong c´c b`i to´n sau dˆy (23-34) h˜y t´nh thˆ t´ch cua vˆt tr`n a a a . .o.c bo.i ph´p quay h`nh ph˘ng D gi´.i han bo.i du.`.ng (c´c xoay thu du . ’ e ı ’ a o . ’ o a du.`.ng) cho tru.´.c xung quanh truc cho tru.´.c o o . o 23. D : y 2 = 2px, x = a; xung quanh truc Ox. (DS. πpa2) .
  • 90. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 89 x2 y 2 4π 224. D : 2 + 2 1 (b < a) xung quanh truc Oy. (DS. . a b) a b 3 x2 y 2 4π 225. D : 2 + 2 1 (b < a) xung quanh truc Ox. (DS. . ab ) a b 3 226. D : 2y = x2 ; 2x + 2y − 3 = 0 xung quanh truc Ox. (DS. 18 π) . 15 2 2 π27. D : x + y = 1; x + y = 1 xung quanh truc Ox. (DS. ) . 3 2 228. D : x + y = 4, x = −1, x = 1, y > 0 xung quanh truc Ox. . (DS. 8π) π229. D : y = sin x, 0 x π, y = 0 xung quanh truc Ox. (DS. . ) 2 x2 y 2 430. D : 2 − 2 = 1, y = 0, y = b xung quanh truc Oy. (DS. πa2b) . a b 3 231. D : y 2 + x − 4 = 0, x = 0 xung quanh truc Oy. (DS. 34 π) . 1532. D : xy = 4, y = 0, x = 1, x = 4 xung quanh truc Ox. (DS. 12π) .33. D : x2 + (y − b)2 R2 (0 < R b) xung quanh truc Ox. . (DS. 2π 2 bR2 ) Chı dˆ n. H` tr`n D c´ thˆ xem nhu. hiˆu cua hai thang cong ’ ˜a ınh o o e’ e ’ . √ D1 = (x, y) : −R x R, 0 y − R2 − x2 v` a √ D2 = (x, y) : −R x R, 0 y + R2 − x2 . √34∗. D = (x, y) : 0 y R2 − x2 xung quanh du.`.ng th˘ng o ’ ay = R. 3π − 4 3 (DS. πR ) 3 ’ ˜ ’ ´ . e ’ e o . o ` e Chı dˆ n. Chuyˆn gˆc toa dˆ vˆ diˆm (0, R). a11.3.2 T´ ınh dˆ d`i cung v` diˆn t´ o a . a e . ıch m˘t tr`n a . o xoay1+ Nˆu du.`.ng cong L(A, B) du.o.c cho bo.i phu.o.ng tr`nh y = y(x), ´ e o . ’ ıx ∈ [a, b] (hay x = g(y)) ho˘c bo.i c´c phu.o.ng tr` tham sˆ x = ϕ(t), a ’ a . ınh ´ o
  • 91. 90 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . y = ψ(t) th` vi phˆn dˆ d`i cung du.o.c biˆu diˆn bo.i cˆng th´.c ı a o a . . e’ ˜ e ’ o u d= 1 + (yx )2 dx = 1 + (xy )2 dy = xt 2 + yt2 dt (11.17) v` dˆ d`i cua du.`.ng cong L(A, B) du.o.c t´nh bo.i cˆng th´.c a o a ’ . o . ı ’ o u xB =b yB (A, B) = 1 + (y )2 dx = 1 + (xy )2 dy xA =a yA tB = xt 2 + yt 2 dt. (11.18) tA Nˆu du.`.ng cong du.o.c cho bo.i phu.o.ng tr` trong toa dˆ cu.c ρ = ρ(ϕ) ´ e o . ’ ınh . o . . th` ı 2 d = ρ2 + ρϕ dϕ v` a ϕB 2 (A, B) = ρ2 + ρϕ dϕ. (11.19) ϕA 2+ Nˆu m˘t σ thu du.o.c do quay du.`.ng cong cho trˆn [a, b] bo.i ´ e a . . o e ’ h`m khˆng ˆm y = f (x) a o a 0 xung quanh truc Ox th` vi phˆn diˆn . ı a e . t´ m˘t ıch a . y + (y + dy) ds = 2π · d = π(2y + dy)d ≈ 2πyd 2 v` diˆn t´ m˘t tr`n xoay du.o.c t´nh theo cˆng th´.c a e ıch a o . . . ı o u b Sx = 2π f (x) 1 + (fx )2 dx. (11.20) a
  • 92. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 91Nˆu quay du.`.ng cong L(A, B) xung quanh truc Oy th` ds ≈ 2πx(y)d e´ o . ıv` a yB Sy = 2π x(y) 1 + (xy )2 dy. (11.21) yANˆu du.`.ng cong L(A, B) du.o.c cho bo.i phu.o.ng tr` tham sˆ x = ϕ(t), ´ e o . ’ ınh ´ oy = ψ(t) 0 (t ∈ [α, β]) th`ı β Sx = 2π ψ(t) ϕ 2 + ψ 2 dt. (11.22) α Tu.o.ng tu. ta c´ . o β Sy = 2π ϕ(t) ϕ 2 + ψ 2 dt, ϕ(t) 0. (11.23) α CAC V´ DU ´ I .V´ du 1. T´ dˆ d`i du.`.ng tr`n b´n k´nh R. ı . ınh o a . o o a ı Giai. Ta c´ thˆ xem du.`.ng tr`n d˜ cho c´ tˆm tai gˆc toa dˆ. ’ o e ’ o o a o a ´ . o . o .Phu .o.ng tr` du.`.ng tr`n du.´.i dang tham sˆ c´ dang x = R cos t, ınh o o o . ´ o o .y = R sin t, t ∈ [0, 2π]. Ta chı cˆn t´nh dˆ d`i cua mˆt phˆn tu. du.`.ng ’ ` ı a o a ’ . o . `a o πtr`n u.ng v´.i 0 t o ´ o l` du. Theo cˆng th´.c (11.18) ta c´ a ’ o u o 2 π/2 π/2 =4 (−R sin t)2 + (R cos t)2 dt = 4Rt = 2πR. 0 0V´ du 2. T´ ı . ınh dˆ d`i cua v`ng th´. nhˆt cua du.`.ng xo˘n ˆc o a ’ . o u ´ ’ a o ´ o a ´Archimedes ρ = aϕ. Giai. Theo dinh ngh˜ du.`.ng xo˘n ˆc Archimedes l` du.`.ng cong ’ . ıa, o ´ ´ a o a oph˘ng vach nˆn bo.i mˆt diˆm chuyˆn dˆng dˆu theo mˆt tia xuˆt ph´t ’ a . e ’ o e . ’ ’ . e o `e o . a´ a
  • 93. 92 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . t`. gˆc-cu.c m` tia n`y lai quay xung quanh gˆc cu.c v´.i vˆn tˆc g´c u o .´ a a . ´ o . . ´ o a o o ´ cˆ dinh. V`ng th´ o . o . nhˆt cua du.`.ng xo˘n ˆc Archimedes du.o.c tao nˆn u a ’´ o ´ o a ´ . . e khi g´c cu.c ϕ biˆn thiˆn t`. 0 dˆn 2π. Do d´ theo cˆng th´.c (11.19) o . e´ e u ´ e o o u ta c´o 2π 2π = a2ϕ2 + a2dϕ = a ϕ2 + 1dϕ. 0 0 T´ phˆn t`.ng phˆn b˘ng c´ch d˘t u = ıch a u ` a a ` a a . ϕ2 + 1, dv = dϕ ta c´ o 2π 2π ϕ2 =a ϕ ϕ2 + 1 − dϕ 0 ϕ2 + 1 0 2π 2π ϕ2 + 1 − 1 =a ϕ ϕ2 + 1 − dϕ 0 ϕ2 + 1 0 1 1 2π = a ϕ ϕ2 + 1 + ln(ϕ + ϕ2 + 1) 2 2 0 √ 1 √ = a π 4π 2 + 1 + 2π + 4π 2 + 1 . 2 ı . ınh e ıch a ` V´ du 3. T´ diˆn t´ m˘t cˆu b´n k´nh R. . . a a ı Giai. C´ thˆ xem m˘t cˆu c´ tˆm tai gˆc toa dˆ v` thu du.o.c bo.i ’ o e ’ a ` o a . a √ ´ . o . o a . . ’ ph´p quay nu.a du.`.ng tr`n y = R2 − x2 xung quanh truc Ox. e ’ o o . Phu .o.ng tr` du.`.ng tr`n c´ dang x2 + y 2 = R2 . Do d´ y = ınh o o o . o x .c (11.20) ta c´ −√ . Theo cˆng th´ o u o R2 − x2 R R √ x2 √ Sx = 2π R2 − x2 · 1+ 2 dx = 2π R2 − x2 + x2 dx R − x2 −R −R R = 2πRx = 4πR2 . −R V´ du 4. T´ diˆn t´ m˘t tao nˆn bo.i ph´p quay du.`.ng lemniscat ı . ınh e ıch a . e . . ’ e o √ ρ = a cos 2ϕ xung quanh truc cu.c. . .
  • 94. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 93 Giai. Biˆn ρ chı nhˆn gi´ tri thu.c khi cos 2ϕ ’ ´ e ’ a. a . . 0 t´.c l` khi u a e ’−π/4 ϕ π/4 (nh´nh bˆn phai) hay khi 3π/4 ϕ a 5π/4 (nh´nh a e a a ’ `bˆn tr´i). Vi phˆn cung cua lemniscat b˘ng a a sin 2ϕ 2 d = ρ2 + ρ 2 dϕ = a2 cos 2ϕ + (− √ dϕ cos 2ϕ adϕ =√ · cos 2ϕ √Ngo`i ra y = ρ sin ϕ = a cos 2ϕ · sin ϕ. T`. d´ diˆn t´ cˆn t` b˘ng a u o e ıch ` ım ` . a a ` e ıch ’ .o.c bo.i ph´p quay nh´nh phai. Do d´hai lˆn diˆn t´ cua m˘t thu du . a a ’ e a ’ o . .theo (11.20) π/4 π/4 √ a cos 2ϕ · sin ϕ · adϕ S = 2 · 2π yds = 4π √ cos 2ϕ 0 0 π/4 √ = 4π a2 sin ϕdϕ = 2πa2(2 − 2). 0V´ du 5. T` diˆn t´ch m˘t tao nˆn bo.i ph´p quay cung parabˆn ı . ım e ı . a . . e ’ e o x2 √y= ,0 x 3 xung quanh truc Oy. . 2 √ 1 Giai. Ta c´ x = 2y, x = √ . Do d´, ´p dung cˆng th´.c ’ o o a . o u 2y(11.18) ta thu du.o.c . 3/2 3/2 1 S = 2π 2y 1 + dy = 2π 2y + 1dy 2y 0 0 (2y + 1)3/2 3/2 14π = 2π · · = 3 0 3V´ du 6. T` diˆn t´ m˘t tao nˆn bo.i ph´p quay elip x2 + 4y 2 = 26 ı . ım e ıch a . e . . ’ exung quanh: a) truc Ox; b) truc Oy. . . Giai. Nu.a trˆn cua elip d˜ cho c´ thˆ xem nhu. dˆ thi cua h`m ’ ’ e ’ a o e ’ ` . ’ o a 1√y= 36 − x2 ; −6 x 6. H`m n`y khˆng c´ dao h`m khi x = ±6, a a o o . a 2
  • 95. 94 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . o e ’ c`n trˆn khoang (−6, 6) dao h`m khˆng bi ch˘n. Do vˆy khˆng thˆ . a o . a. a . o ’ e ınh ` .c (11.20) trong toa dˆ Dˆ c´c du.o.c. t´ b˘ng cˆng th´ a o u . o ` a . e . ’ kh˘c phuc kh´ kh˘n d´, ta d`ng ph´p tham sˆ h´a du.`.ng elip: Dˆ a e ´ . o a o u e ´ o o o x = 6 cos t, y = 3 sin t, 0 t 2π. 1+ Ph´p quay xung quanh truc Ox. Ta x´t nu.a trˆn cua elip tu.o.ng e . e ’ e ’ u.ng v´.i 0 t π. Theo cˆng th´.c (11.22) du.´.i dang tham sˆ ta c´ ´ o o u o . o´ o π Sx = 2π 3 sin t · 36 sin2 t + 9 cos2 tdt. 0 2 D˘t cos t = √ sin ϕ ta c´ a . o 3 π/3 √ √ √ Sx = 24 3π cos2 ϕdϕ = 2 3π(4π + 3 3). −π/3 2+ Ph´p quay xung quanh truc Oy. Ta x´t nu.a bˆn phai cua elip e . e ’ e ’ ’ π π (tu.o.ng u.ng v´.i t ∈ − , . Tu.o.ng tu. nhu. trˆn ta ´p dung (11.23) ´ o . e a . .o.c 2 2 v` thu du . a π/2 1 Sy = 2π 6 cos t · 36 sin2 t + 9 cos2 tdt D˘t sin t = √ shϕ a . 3 −π/2 √ arcsh 3 √ √ √ √ = 24 3π ch2ϕdϕ = 24 3π 2 3 + ln(2 + 3) . √ −arcsh 3 ` ˆ BAI TAP . T´ dˆ d`i cung cua du.`.ng cong ınh o a . ’ o 8 √ 1. y = x3/2 t`. x = 0 dˆn x = 4. (DS. u ´ e (10 10 − 1)) 27
  • 96. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 95 √ 1 √2. y = x2 − 1 t`. x = −1 dˆn x = 1. (DS. u ´ e 5 + ln(2 + 5)) 2 a a(e2 − 1)3. y = ex/a + e−x/a t`. x = 0 dˆn x = a. (DS. u ´ e ) 2 2e π 14. y = ln cos x t`. x = 0 dˆn x = . (DS. ln 3) u ´ e 6 2 π 2π5. y = ln sin x t`. x = dˆn x = u ´ e . (DS. ln 3) 3 3 π √6. x = et sin t, y = et cos t, 0 t . (DS. 2(eπ/2 − 1)) 27. x = a(t − sin t), y = a(1 − cos t); 0 t 2π. (DS. 8a)8. x = a cos3 t, y = a sin3 t; 0 t 2π. (DS. 6a) 3a ’ a˜ Chı dˆ n. V` ı xt 2 + yt2 = | sin 2t| v` h`m | sin 2t| c´ chu k` π/2 a a o y 1 π/2nˆn e =4 d . 0 √9. x = et cos t, y = et sin t t`. t = 0 dˆn t = ln π. (DS. u ´ e 2(π − 1)) π10. x = 8 sin t + 6 cos t, y = 6 sin t − 8 cos t t`. t = 0 dˆn t = . (DS. u ´ e 25π)11. ρ = aekθ (du.`.ng xo˘n ˆc lˆga) t`. θ = 0 dˆn θ = T . o ´ ´ a o o u ´ e a√ 2 (ekT − 1)) (DS. 1+k k12. ρ = a(1 − cos ϕ), a > 0, 0 ϕ 2π (du.`.ng h`nh tim). (DS. 8a) o ı 1 113∗. ρϕ = 1 t`. diˆm A 2, u e ’ ´ ’ dˆn diˆm B e e , 2 - du.`.ng xo˘n ˆc o ´ o a ´ 2 2hypecbon. √ √ 5 3+ 5 (DS. + ln ) 2 2 T´ diˆn t´ c´c m˘t tr`n xoay thu du.o.c khi quay cung du.`.ng ınh e ıch a . a . o . o .`.ng cong xung quanh truc cho tru.´.c.cong hay du o o . 2 214. Cung cua du.`.ng y = x3 t`. x = − dˆn x = xung quang truc ’ o u ´ e . 3 3Ox.
  • 97. 96 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 2π 125 (DS. −1 ) 27 27 15. Du.`.ng x = a cos3 t, y = a sin3 t xung quanh truc Ox. o . 12 2 (DS. πa ) 5 x2 y 2 16. 2 + 2 = 1, a > b xung quanh truc Ox. . a b a ’ (DS. 2πb b + arc sin ε , ε l` tˆm sai cua elip) a a ε Chı dˆ n. Dao h`m hai vˆ phu.o.ng tr`nh elip rˆi r´t ra yy = ’ a ˜ . a ´ e ı ` u o 2 bx − 2 , c`n biˆu th´.c du.´.i dˆu t´ch phˆn du.o.c viˆt y 1 + y 2dx = o ’ e u ´ o a ı a . ´ e a y 2 + (yy )3 dx. 17. Cung du.`.ng tr`n x2 + (y − b)2 = R (khˆng c˘t truc Oy) t`. y1 o o o ´ a . u ´ dˆn y2 xung quanh truc Oy. (DS. 2πR(y2 − y1)) e . Chı dˆ n. M˘t thu du.o.c l` d´.i cˆu. ’ ˜a a . . a o ` a 18. y = sin x t`. x = 0 dˆn x = π xung quanh truc Ox. u ´ e . √ √ (DS. 2π 2 + ln(1 + 2) ) x3 . 19. y = u ´ t` x = −2 dˆn x = 2 xung quanh truc Ox. e . 3 √ 34 17 − 2 (DS. π) 9 20. Cung bˆn tr´i du.`.ng th˘ng x = 2 cua du.`.ng cong y 2 = 4 + x, e a o ’ a ’ o 62π xung quanh truc Ox. (DS. . ) 3 a 21. y = ex/a + e−x/a t`. x = 0 dˆn x = a (a > 0). u ´ e 2 πa2 2 (DS. (e + 4 − e−2 )) 4 56π 22. y 2 = 4x t`. x = 0 dˆn x = 3, xung quanh truc Ox. (DS. u ´ e . ) 3 π 23. x = et sin t, y = et cos t t`. t = 0 dˆn t = , xung quanh truc Ox. u ´ e . 2 √ 2π 2 π (DS. (e − 2)) 5
  • 98. 11.3. Mˆt sˆ u.ng dung cua t´ phˆn x´c d .nh . ´ o o´ . ’ ıch a a i 9724. x = a cos3 t, y = a sin3 t, 0 t 2π; quay xung quanh truc Ox. . 12 2 (DS. πa ) 5 Chı dˆ n. V` du.`.ng cong c´ t´ dˆi x´.ng qua c´c truc toa dˆ nˆn ’ ˜ a ı o ´ o ınh o u a . . o e. ’ ` ıchı cˆn t´nh diˆn t´ tao nˆn bo a e ıch . e ’.i mˆt phˆn tu. du.`.ng thuˆc g´c I o ` a o o o . . .quay xung quanh truc Ox. .25. x = t − sin t, y = 1 − cos t (diˆn t´ du.o.c tao th`nh khi quay e ıch . . . amˆt cung); xung quanh truc Ox. o. . 64π (DS. ) 3 π26. y = sin 2x t`. x = 0 dˆn x = ; xung quanh truc Ox. u ´ e . 2 π √ √ (DS. 2 5 + ln( 5 + 2) ) 2 2 227. 3x + 4y = 12; xung quanh truc Oy. (DS. 2π(4 + 3 ln 3)) . 62π28. x2 = y + 4, y = 2; xung quanh truc Oy. (DS. . ) 329. Cung cua du.`.ng tr`n x2 + y 2 = 4 (y > 0) gi˜.a hai diˆm c´ ho`nh ’ o o u e’ o adˆ x = −1 v` x = 1; xung quanh truc Ox. (DS. 8π) o . a .30. Du.`.ng h` tim (cacdiod) ρ = a(1 + cos ϕ); quay xung quanh o ınhtruc cu.c. . . 32πa2 (DS. ) 531. Du.o.ng tr`n ρ = 2r sin ϕ; quay xung quanh truc cu.c. (DS. 4π 2 r2 ) ` o . .32. Cung AB cua du.`.ng xicloid x = a(t − sin t), y = a(1 − cos t); ’ o √ πa2quay xung quanh du.`.ng th˘ng y = a. (DS. 16 2 o ’ a ) 3 Chı dˆ n. Ap dung cˆng th´.c ’ ˜ ´ a . o u 3π S = 2π 2(y(t) − a) xt2 + yt 2 dt. π/2
  • 99. 98 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 11.4 T´ phˆn suy rˆng ıch a o . 11.4.1 T´ phˆn suy rˆng cˆn vˆ han ıch a o . a . o . 1. Gia su. h`m f(x) x´c dinh ∀ x ’ ’ a a . a ’ ı a v` kha t´ch trˆn moi doan [a, b]. e . . ´ o e ` . o .i han h˜.u han Nˆu tˆn tai gi´ . u . b lim f (x)dx (11.24) b→+∞ a th` gi´.i han d´ du.o.c goi l` t´ phˆn suy rˆng cua h`m f (x) trˆn ı o . o . . a ıch a o . ’ a e +∞ ’ khoang [a, +∞) v` k´ hiˆu l` a y e a. f (x)dx. a Trong tru.`.ng ho.p n`y ngu.`.i ta c`n n´i r˘ng t´ phˆn suy rˆng o . a o o o a ` ıch a o . . ’ ı ı o . e ’ (11.24) hˆi tu v` h`m f(x) kha t´ch theo ngh˜a suy rˆng trˆn khoang o . a a +∞ [a, +∞). Nˆu gi´.i han (11.24) khˆng tˆn tai th` t´ch phˆn ´ e o . o ` . o ı ı a f (x)dx a du.o.c goi l` t´ phˆn phˆn k` v` h`m f (x) khˆng kha t´ch theo ngh˜ . . a ıch a a y a a o ’ ı ıa suy rˆng trˆn [a, +∞). o. e Tu .o.ng tu. nhu. trˆn, theo dinh ngh˜a e ı . . b b f(x)dx = lim f (x)dx (11.25) a→−∞ −∞ a +∞ c +∞ f(x)dx = f (x)dx + f (x)dx, c ∈ R. (11.26) −∞ −∞ c 2. C´c cˆng th´.c co. ban dˆi v´.i t´ phˆn suy rˆng a o u ’ ´ o o ıch a o . +∞ ´ ´ 1) T´ tuyˆn t´nh. Nˆu c´c t´ch phˆn suy rˆng ınh e ı e a ı a o . f (x)dx v` a a +∞ +∞ g(x)dx hˆi tu ∀ α, β ∈ R th` t´ch phˆn o . . ı ı a (αf (x) + βg(x))dx hˆi tu o . . a a
  • 100. 11.4. T´ phˆn suy rˆng ıch a o . 99v` a +∞ +∞ +∞ (αf (x) + βg(x))dx = α f (x)dx + β g(x)dx. a a a 2) Cˆng th´.c Newton-Leibnitz. Nˆu trˆn khoang [a, +∞) h`m f (x) o u ´ e e ’ a e . a a e a a o ’ o ıliˆn tuc v` F (x), x ∈ [a, +∞) l` nguyˆn h`m n`o d´ cua n´ th` +∞ +∞ f(x)dx = F (x) a = F (+∞) − F (a) atrong d´ F (+∞) = lim F (x). o x→+∞ 3) Cˆng th´.c dˆi biˆn. Gia su. f(x), x ∈ [a, +∞) l` h`m liˆn tuc, o u o e ’ ´ ’ ’ a a e . a ’ϕ(t), t ∈ [α, β] l` kha vi liˆn tuc v` a = ϕ(α) ϕ(t) < lim ϕ(t) = e . a t→β−0+∞. Khi d´: o +∞ β f(x)dx = f (ϕ(t))ϕ (t)dt. (11.27) a α 4) Cˆng th´.c t´ch phˆn t`.ng phˆn. Nˆu u(x) v` v(x), x ∈ [a, +∞) o u ı a u ` a ´ e al` nh˜.ng h`m kha vi liˆn tuc v` lim (uv) tˆn tai th` a u a ’ e . a ` . o ı: x→+∞ +∞ +∞ +∞ udv = uv a − vdu (11.28) a a +∞trong d´ uv o a = lim (uv) − u(a)v(a). x→+∞ a ` 3. C´c diˆu kiˆn hˆi tu e e . o . . +∞ ’ 1) Tiˆu chuˆn Cauchy. T´ch phˆn e a ı a a ’ f (x)dx hˆi tu khi v` chı khi o . . a∀ ε > 0, ∃ b = b(ε) a sao cho ∀ b1 > b v` ∀ b2 > b ta c´: a o b2 f(x)dx < ε. b1
  • 101. 100 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 2) Dˆu hiˆu so s´nh I. Gia su. g(x) f (x) 0 ∀ x ´ a e . a ’ ’ a v` f (x), a ’ ıch e g(x) kha t´ trˆn moi doan [a, b], b < +∞. Khi d´: . . o +∞ +∞ ´ (i) Nˆu t´ phˆn e ıch a g(x)dx hˆi tu th` t´ch phˆn o . ı ı . a f (x)dx hˆi tu. o . . a a +∞ +∞ ´ (ii) Nˆu t´ phˆn e ıch a f (x)dx phˆn k` th` t´ phˆn a y ı ıch a g(x)dx phˆn a a a k`. y 3) Dˆu hiˆu so s´nh II. Gia su. f(x) ´ a e . a ’ ’ 0, g(x) > 0 ∀ x a v` a f (x) lim = λ. x→+∞ g(x) Khi d´: o +∞ +∞ ´ (i) Nˆu 0 < λ < +∞ th` c´c t´ch phˆn e ı a ı a f (x)dx v` a g(x)dx a a dˆng th`.i hˆi tu ho˘c dˆng th`.i phˆn k`. ` o o o . a ` . . o o a y +∞ ´ (ii) Nˆu λ = 0 v` t´ch phˆn e a ı a g(x)dx hˆi tu th` t´ch phˆn o . . ı ı a a +∞ f(x)dx hˆi tu. o . . a +∞ ´ (iii) Nˆu λ = +∞ v` t´ phˆn e a ıch a f (x)dx hˆi tu th` t´ch phˆn o . ı ı . a a +∞ g(x)dx hˆi tu. o . . a Dˆ so s´nh ta thu.`.ng su. dung t´ phˆn ’ e a o ’ . ıch a +∞ dx o . e . ´ hˆi tu nˆu α > 1, (11.29) xα ´ phˆn k` nˆu α a y e 1. a
  • 102. 11.4. T´ phˆn suy rˆng ıch a o . 101 +∞-.Dinh ngh˜ T´ phˆn ıa. ıch a f(x)dx du.o.c goi l` hˆi tu tuyˆt dˆi nˆu . . a o .. . ´ ´ e o e a +∞t´ phˆn ıch a |f(x)|dx hˆi tu v` du.o.c goi l` hˆi tu c´ diˆu kiˆn nˆu o . a . . . a o . o ` . e e e . ´ a +∞ +∞t´ phˆn ıch a f(x)dx hˆi tu nhu.ng t´ phˆn o . . ıch a |f (x)|dx phˆn k`. a y a a . ıch a o . . e o ` o . . ´ e Moi t´ phˆn hˆi tu tuyˆt dˆi dˆu hˆi tu. . . dˆu hiˆu so s´nh II v` (11.29) r´t ra u ´ 3) T` a e a a u . Dˆu hiˆu thu.c h`nh. Nˆu khi x → +∞ h`m du.o.ng f(x) l` vˆ ´ a e . . a e´ a a o 1c`ng b´ cˆp α > 0 so v´.i th` u e a´ o ı x +∞ (i) t´ch phˆn ı a f(x)dx hˆi tu khi α > 1; o . . a +∞ (ii) t´ phˆn ıch a f(x)dx phˆn k` khi α a y 1. a CAC V´ DU ´ I .V´ du 1. T´ t´ phˆn ı . ınh ıch a +∞ dx I= √ · x2 x2 − 1 2 ’ Giai. Theo dinh ngh˜ ta c´ . ıa o +∞ b dx dx √ = lim √ · x 2 x2 − 1 b→+∞ x2 x2 − 1 2 2 1D˘t x = , ta thu du.o.c a . . t
  • 103. 102 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . b 1/b 1/b dx −dt tdt I(b) = √ = =− √ x 2 x2 − 1 1 1 1 − t2 2 1/2 t2 · −1 1/2 t2 t2 √ 1/b 1 1 = 1 − t2 = 1− 2 − 1− . 1/2 b 4 √ 2− 3 T`. d´ suy r˘ng I = lim I(b) = u o ` a . Nhu. vˆy t´ phˆn d˜ cho a ıch a a . b→+∞ 2 hˆi tu. o . . V´ du 2. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a +∞ 2x2 + 1 dx. x3 + 3x + 4 1 Giai. H`m du.´.i dˆu t´ phˆn > 0 ∀ x ’ a ´ o a ıch a 1. Ta c´ o 1 2x2 + 1 2+ f(x) = 3 = · x2 x + 3x + 4 3 4 x+ + 2 x x 2 V´.i x du l´.n h`m f(x) c´ d´ng diˆu nhu. . Do d´ ta lˆy h`m o ’ o a o a e . o ´ a a x 1 ’ ϕ(x) = dˆ so s´nh v` c´ e a a o x f(x) (2x2 + 1)x lim = lim 2 = 2 = 0. x→+∞ ϕ(x) x→+∞ x + 3x + 4 ∞ dx V` t´ phˆn ı ıch a ´ phˆn k` nˆn theo dˆu hiˆu so s´nh II t´ch phˆn d˜ a y e a e . a ı a a x 1 cho phˆn k`. a y V´ du 3. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a ∞ dx √ 3 · x3 − 12 2
  • 104. 11.4. T´ phˆn suy rˆng ıch a o . 103 Giai. Ta c´ bˆt d˘ng th´.c ’ ´ ’ o a a u 1 1 √ 3 > khi x > 2. x3 −1 x ∞ dxNhu.ng t´ phˆn ıch a o ´ phˆn k`, do d´ theo dˆu hiˆu so s´nh I t´ch a y a e . a ı x 2phˆn d˜ cho phˆn k`. a a a yV´ du 4. Khao s´t su. hˆi tu v` d˘c t´nh hˆi tu cua t´ch phˆn ı . ’ a . o . a a ı . . o . ’ ı . a +∞ sin x dx. x 1 Giai. Dˆu tiˆn ta t´ phˆn t`.ng phˆn mˆt c´ch h`nh th´.c ’ ` a e ıch a u ` a o a . ı u +∞ +∞ +∞ sin x cos x +∞ cos x cos x dx = − − dx = cos 1 − dx. x x 1 x2 x2 1 1 1 (11.30) +∞ cos x T´ phˆn ıch a dx hˆi tu tuyˆt dˆi, do d´ n´ hˆi tu. Nhu. vˆy o . . . ´ e o o o o . . a . x2 1ca hai sˆ hang o. vˆ phai (11.30) h˜.u han. T`. d´ suy ra ph´p t´ ’ ´ o . ’ e´ ’ u . u o e ıchphˆn t` a u .ng phˆn d˜ thu.c hiˆn l` ho.p l´ v` vˆ tr´i cua (11.30) l` t´ch ` a a . e a . y a e a ’ ´ a ı .phˆn hˆi tu. a o . . Ta x´t su. hˆi tu tuyˆt dˆi. Ta c´ e . o . . . ´ e o o 1 − cos 2x | sin x| sin2 x = 2v` do vˆy ∀ b > 1 ta c´ a a . o b b b | sin x| 1 dx 1 cos 2x dx − dx. (11.31) x 2 x 2 x 1 1 1
  • 105. 104 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . T´ phˆn th´. nhˆt o. vˆ phai cua (11.31) phˆn k`. T´ch phˆn th´. ıch a u ´ a ’ e ´ ’ ’ a y ı a u . vˆ phai d´ hˆi tu (diˆu d´ du.o.c suy ra b˘ng c´ch t´ch phˆn t`.ng ’ ´ hai o e ’ o o . ` o e ` a a ı a u . . phˆn nhu. (11.30)). Qua gi´.i han (11.31) khi b → +∞ ta c´ vˆ phai ` a o . o e ´ ’ ’ ` a e ´ a o ıch a e a ’ ´ cua (11.31) dˆn dˆn ∞ v` do d´ t´ phˆn vˆ tr´i cua (11.31) phˆn a .c l` t´ phˆn d˜ cho hˆi tu c´ diˆu kiˆn (khˆng tuyˆt dˆi). o . o ` k`, t´ a ıch a a y u . e e . o . ´ e o ` ˆ BAI TAP . T´ c´c t´ phˆn suy rˆng cˆn vˆ han ınh a ıch a o . a o . . ∞ 2 1 1. xe−x dx (DS. ) 2 0 ∞ dx π 2. √ . (DS. ) x x2 − 1 6 0 ∞ dx π−2 3. . (DS. ) (x2 + 1)2 8 0 ∞ 4. x sin xdx. (DS. Phˆn k`) a y 0 ∞ 2xdx 5. . (DS. Phˆn k`) a y x2 + 1 −∞ ∞ 1 6. e−x sin xdx. (DS. ) 2 0 +∞ 1 2 2 1 7. + dx. (DS. + ln 3) x2 − 1 (x + 1)2 3 2 2 +∞ √ dx π 5 8. . (DS. ) x2 + 4x + 9 5 −∞
  • 106. 11.4. T´ phˆn suy rˆng ıch a o . 105 +∞ xdx 1 √9. . (DS. ’ ˜ ). Chı dˆ n. D˘t x = t. a a . √ (x2 + 1)3 36 2 +∞ dx 2 110. √ . ’ a˜ (DS. ln 1 + √ ). Chı dˆ n. D˘t x = . a . x x2 + x + 1 3 t 1 +∞ arctgx π ln 211. dx. (DS. + ) x2 4 2 1 +∞ 2x + 512. dx. (DS. Phˆn k`) a y x2 + 3x − 10 3 ∞ b13. e−ax sin bxdx, a > 0. (DS. ) a2 + b2 0 +∞ a14. e−ax cos bxdx, a > 0. (DS. ) a2 + b2 0 Khao s´t su. hˆi tu cua c´c t´ phˆn suy rˆng cˆn vˆ han ’ a . o . ’ a ıch a . o . a o . . ∞ e−x15. dx. (DS. Hˆi tu) o . . x 1 e−x Chı dˆ n. Ap dung bˆt d˘ng th´.c ’ a ´ ˜ . ´ ’ a a u e−x ∀ x 1. x +∞ xdx16. √ . (DS. Phˆn k`) a y x4 + 1 2 Chı dˆ n. Ap dung bˆt d˘ng th´.c ’ ˜ ´ a . ´ ’ a a u x x √ >√ ∀x 2. x4 + 1 x4 + x4 +∞ sin2 3x17. √ 3 dx. (DS. Hˆi tu) o . . x4 + 1 1
  • 107. 106 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . +∞ dx 18. √ . (DS. Phˆn k`) a y 4x + ln x 1 +∞ 1 ln 1 + 19. x dx. ´ (DS. Hˆi tu nˆu α > 0) o . e xα . 1 +∞ xdx 20. √ 3 . (DS. Hˆi tu) o . . x5 + 2 0 +∞ cos 5x − cos 7x 21. dx. (DS. Hˆi tu) o . . x2 1 +∞ xdx 22. √ 3 . (DS. Hˆi tu) o . . 1 + x7 0 +∞ √ x+1 23. √ dx. (DS. Hˆi tu) o . . 1 + 2 x + x2 0 ∞ 1 24. √ (e1/x − 1)dx. (DS. Hˆi tu) o . . x 1 ∞ √ x+ x+1 25. √ dx. (DS. Phˆn k`) a y x2 + 2 5 x4 + 1 1 ∞ dx 26. . (DS. Hˆi tu) o . . x(x − 1)(x − 2) 3 ∞ 2 ∗ 27 . (3x4 − x2)e−x dx. (DS. Hˆi tu) o . . 0 +∞ x2 Chı dˆ n. So s´nh v´.i t´ phˆn hˆi tu ’ ˜a a o ıch a o . . e− 2 dx (tai sao ?) v` ´p . aa 0 . ´ dung dˆu hiˆu so s´nh II. a e . a
  • 108. 11.4. T´ phˆn suy rˆng ıch a o . 107 +∞ ∗ ln(x − 2)28 . dx. (DS. Hˆi tu) o . . x5 + x2 + 1 5 Chı dˆ n. Ap dunng hˆ th´.c ’ a ´ ˜ . e u . ln t ln(x − 2) lim = 0 ∀α > 0 ⇒ lim = 0 ∀ α > 0. t→+∞ tα x→+∞ xα +∞ dxT`. d´ so s´nh t´ phˆn d˜ cho v´.i t´ phˆn hˆi tu u o a ıch a a o ıch a o . . , α > 1. xα 5 ´ ´ . ´Tiˆp dˆn ´p dung dˆu hiˆu so s´nh II. e e a a e . a11.4.2 o . ’ T´ phˆn suy rˆng cua h`m khˆng bi ch˘n ıch a a o . a .1. Gia su. h`m f (x) x´c dinh trˆn khoang [a, b) v` kha t´ch trˆn moi ’ ’ a a . e ’ a ’ ı e . ´ o e ` . o .i han h˜.u handoan [a, ξ], ξ < b. Nˆu tˆn tai gi´ . u . . ξ lim f (x)dx (11.32) ξ→b−0 0th` gi´.i han d´ du.o.c goi l` t´ phˆn suy rˆng cua h`m f (x) trˆn [a, b) ı o . o . . a ıch a o . ’ a ev` k´ hiˆu l`: a y e a . b f (x)dx. (11.33) a Trong tru.`.ng ho.p n`y t´ phˆn suy rˆng (11.33) du.o.c goi l` t´ o . a ıch a o . . . a ıch ´phˆn hˆi tu. Nˆu gi´ . a o . e o.i han (11.32) khˆng tˆn tai th` t´ch phˆn suy o `o . ı ı a .rˆng (11.33) phˆn k`. o . a y . o. ’ a Dinh ngh˜ t´ phˆn suy rˆng cua h`m f (x) x´c dinh trˆn khoang ıa ıch a a . e ’ .o.c ph´t biˆu tu.o.ng tu..(a, b] du . a e’ . ´ e a ’ ıch Nˆu h`m f (x) kha t´ theo ngh˜a suy rˆng trˆn c´c khoang [a, c) ı o . e a ’v` (c, b] th` h`m du.o.c goi l` h`m kha t´ch theo ngh˜a suy rˆng trˆn a ı a . . a a ’ ı ı o . e
  • 109. 108 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . doa n [a, b] v` trong tru.o.ng ho.p n`y t´ch phˆn suy rˆng du.o.c x´c dinh . a ` . a ı a o . . a . .i d˘ng th´.c: ’ ’ bo a u b c b f(x)dx = f (x)dx + f (x)dx. a a c 2. C´c cˆng th´.c co. ban a o u ’ b b ´ 1) Nˆu c´c t´ phˆn e a ıch a f (x)dx v` a g(x)dx hˆi tu th` ∀ α, β ∈ R o . ı . a a ta c´ t´ phˆn o ıch a b [αf (x) + βg(x)]dx hˆi tu v` o . a . a b b b [αf (x) + βg(x)]dx = α f (x)dx + β g(x)dx. a a a 2) Cˆng th´.c Newton-Leibnitz. Nˆu h`m f (x), x ∈ [a, b) liˆn tuc o u ´ e a e . a a o. e a a o ’ v` F (x) l` mˆt nguyˆn h`m n`o d´ cua f trˆn [a, b) th`: e ı b b−0 f(x)dx = F (x) a = F (b − 0) − F (a), a F (b − 0) = lim F (x). x→b−0 3) Cˆng th´.c dˆi biˆn. Gia su. f(x) liˆn tuc trˆn [a, b) c`n ϕ(t), o u o e ’ ´ ’ ’ e . e o ’ t ∈ [α, β) kha vi liˆn tuc v` a = ϕ(α) ϕ(t) < lim ϕ(t) = b. Khi e . a t→β−0 d´: o b β f (x)dx = f [ϕ(t)]ϕ (t)dt. a α
  • 110. 11.4. T´ phˆn suy rˆng ıch a o . 109 4) Cˆng th´.c t´ phˆn t`.ng phˆn. Gia su. u(x), x ∈ [a, b) v` v(x), o u ıch a u ` a ’ ’ ax ∈ [a, b) l` nh˜ a u .ng h`m kha vi liˆn tuc v` lim (uv) tˆn tai. Khi d´; a ’ e . a ` . o o x→b−0 b b b udv = uv a − vdu a a b uv a = lim (uv) − u(a)v(a). x→b−0 ` 3. C´c diˆu kiˆn hˆi tu a e e o . . . 1) Tiˆu chuˆn Cauchy. Gia su. h`m f(x) x´c dinh trˆn khoang e ’ a ’ ’ a a . e ’[a, b), kha t´ theo ngh˜ thˆng thu.`.ng trˆn moi doan [a, ξ], ξ < b ’ ıch ıa o o e . . a o . a . a a. e a ’ e ’v` khˆng bi ch˘n trong lˆn cˆn bˆn tr´i cua diˆm x = b. Khi d´ o bt´ phˆn ıch a a ’ f(x)dx hˆi tu khi v` chı khi ∀ ε > 0, ∃ η ∈ [a, b) sao cho o . . a∀ η1, η2 ∈ (η, b) th` ı η2 f(x)dx < ε. η1 2) Dˆu hiˆu so s´nh I. Gia su. g(x) f (x) 0 trˆn khoang [a, b) ´ a e . a ’ ’ e ’ a ’ ıch e ˜v` kha t´ trˆn mˆ i doan con [a, ξ], ξ < b. Khi d´: o . o b b ´ (i) Nˆu t´ phˆn e ıch a g(x)dx hˆi tu th` t´ch phˆn o . ı ı . a f (x)dx hˆi tu. o . . a a b b ´ (ii) Nˆu t´ phˆn e ıch a f(x)dx phˆn k` th` t´ch phˆn a y ı ı a g(x)dx phˆn a a ak`. y 3) Dˆu hiˆu so s´nh II. Gia su. f(x) ´ a e . a ’ ’ 0, g(x) > 0, x ∈ [a, b) v` a f (x) lim = λ. x→b−0 g(x) Khi d´: o
  • 111. 110 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . b b ´ (i) Nˆu 0 < λ < +∞ th` c´c t´ch phˆn e ı a ı a f (x)dx v` a ` g(x)dx dˆng o a a th`.i hˆi tu ho˘c dˆng th`.i phˆn k`. o o . a ` . . o o a y b b ´ (ii) Nˆu λ = 0 v` t´ phˆn e a ıch a g(x)dx hˆi tu th` t´ phˆn o . ı ıch a . f (x)dx a a hˆi tu. o . . b ´ (iii) Nˆu λ = +∞ v` t´ phˆn e a ıch a f (x)dx hˆi tu th` t´ phˆn o . . ı ıch a a b g(x)dx hˆi tu. o . . a Dˆ so s´nh ta thu.`.ng su. dung t´ phˆn: ’ e a o ’ . ıch a b dx o . e . ´ hˆi tu nˆu α < 1 (b − x)α ´ phˆn k` nˆu α a y e 1 a ho˘c a . b o . e . ´ hˆi tu nˆu α < 1 dx (x − a)α ´ phˆn k` nˆu α a y e 1. a b -. Dinh ngh˜ T´ phˆn ıa. ıch a f (x)dx du.o.c goi l` hˆi tu tuyˆt dˆi nˆu . . a o .. . ´ ´ e o e a b t´ phˆn ıch a |f(x)|dx hˆi tu v` du.o.c goi l` hˆi tu c´ diˆu kiˆn nˆu t´ o . a . . . a o . o ` . e ´ e e ıch . a b b phˆn a f(x)dx hˆi tu nhu.ng o . . |f (x)|dx phˆn k`. a y a a 4) Tu.o.ng tu. nhu. trong 11.4.1 ta c´ . o
  • 112. 11.4. T´ phˆn suy rˆng ıch a o . 111 Dˆu hiˆu thu.c h`nh. Nˆu khi x → b − 0 h`m f (x) 0 x´c dinh ´ a e . . a ´ e a a . 1v` liˆn tuc trong [a, b) l` vˆ c`ng l´.n cˆp α so v´.i a e . a o u o a ´ o th` ı b−x b (i) t´ch phˆn ı a f(x)dx hˆi tu khi α < 1; o . . a b (ii) t´ phˆn ıch a f(x)dx phˆn k` khi α a y 1. a CAC V´ DU ´ I . 1 dxV´ du 1. X´t t´ phˆn ı . e ıch a √ . 1 − x2 0 1 ’ Giai. H`m f(x) = √ a o o ’ ı liˆn tuc v` do d´ n´ kha t´ch trˆn moi e . a e . 1 − x2doan [0, 1 − ε], ε > 0, nhu.ng khi x → 1 − 0 th` f (x) → +∞. Ta c´ ı o . 1−ε dx π lim √ = lim arc sin(1 − ε) = asrc sin 1 = · ε→0 1 − x2 ε→0 2 0Nhu. vˆy t´ phˆn d˜ cho hˆi tu. a ıch a a . o . .V´ du 2. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a 1 √ xdx √ · 1 − x4 0 Giai. H`m du.´.i dˆu t´ phˆn c´ gi´n doan vˆ c`ng tai diˆm ’ a ´ o a ıch a o a . o u . ’ ex = 1. Ta c´ o √ x 1 √ √ ∀ x ∈ [0, 1). 1−x 4 1−x 1 dx Nhu.ng t´ phˆn ıch a √ o . e . ´ hˆi tu, nˆn theo dˆu hiˆu so s´nh I a e . a 1−x 0t´ phˆn d˜ cho hˆi tu. ıch a a o . .
  • 113. 112 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . V´ du 3. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a 1 dx · ex − cos x 0 ’ ’. a a Giai. O dˆy h`m du.´.i dˆu t´ch phˆn c´ gi´n doan vˆ c`ng tai o a ı´ a o a o u . . ’m x = 0. Khi x ∈ (0, 1] ta c´ diˆ e o 1 1 ex − cos x xe 1 1 ı ` v` r˘ng xe a e − cos x (tai sao ?). Nhu.ng t´ch phˆn x . ı a dx phˆn k` a y xe 0 nˆn t´ phˆn d˜ cho phˆn k`. e ıch a a a y V´ du 4. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a +∞ arctgx dx, α 0. xα 0 Giai. Ta chia khoang lˆy t´ phˆn l`m hai sao cho khoang th´. ’ ’ ´ a ıch a a ’ u nhˆt h`m c´ bˆt thu.`.ng tai diˆm x = 0. Ch˘ng han ta chia th`nh hai ´ a a o a´ o . e ’ ’ a . a ’ nu.a khoang (0, 1] v` [1, +∞). Khi d´ ta c´ ’ a o o +∞ 1 +∞ arctgx arctgx arctgx dx = dx + dx. (11.34) xα xα xα 0 0 0 1 arctgx ` Dˆu tiˆn x´t t´ phˆn a e e ıch a dx, Ta c´ o xα 0 arctgx x 1 f (x) = α ∼ α = α−1 = ϕ(x) x (x→0) x x
  • 114. 11.4. T´ phˆn suy rˆng ıch a o . 113 1 T´ phˆn ıch a ϕ(x)dx hˆi tu khi α − 1 < 1 ⇒ α < 2. Do d´ t´ o . . o ıch 0 1phˆn a o . . ´ f(x)dx c˜ng hˆi tu khi α < 2 theo dˆu hiˆu so s´nh II. u a e . a 0 ∞ X´t t´ phˆn e ıch a ´ f(x)dx. Ap dung dˆu hiˆu so s´nh II trong 1◦ ta . ´ a e . a 1 1d˘t ϕ(x) = α v` c´ a . a o x f(x) xαarctgx π lim = lim α = · x→+∞ ϕ(x) x→+∞ x 2 ∞ dxV` t´ phˆn ı ıch a hˆi tu khi α > 1 nˆn v´.i α > 1 t´ phˆn du.o.c o e o ıch a xα . . . 0x´t hˆi tu. Nhu. vˆy ca hai t´ phˆn o. vˆ phai (11.34) chı hˆi tu khi e o . . a ’ . ıch a ’ e ’´ ’ o . .1 < α < 2. o ınh a ` e e o . ’ ı D´ ch´ l` diˆu kiˆn hˆi tu cua t´ch phˆn d˜ cho. . . a aV´ du 5. Khao s´t su. hˆi tu cua t´ch phˆn ı . ’ a . o . ’ ı . a 1 √ 3 ln(1 + x2) √ √ dx. x sin x 0 Giai. H`m du.´.i dˆu t´ phˆn khˆng bi ch˘n trong lˆn cˆn phai ’ a ´ o a ıch a o . a . a a . ’ ’ ’cua diˆm x = 0. Khi x → 0 + 0 ta c´ e o √ 3 √3 ln(1 + x2) x2 1 √ √ ∼ = √ = ϕ(x). x sin x (x→0+0) x 3 x 1 dx V` t´ phˆn ı ıch a o e ´ √ hˆi tu nˆn theo dˆu hiˆu so s´nh II, t´ch phˆn a e a ı a 3 x . . . 0d˜ cho hˆi tu. a o . .
  • 115. 114 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . ` ˆ BAI TAP . T´ c´c t´ phˆn suy rˆng sau. ınh a ıch a o . 6 dx √ 1. . (DS. 6 3 2) 3 (4 − x)2 2 2 dx 2. . (DS. 6) 3 (x − 1)2 0 e dx 3. . (DS. Phˆn k`) a y x ln x 1 2 dx 4. . (DS. Phˆn k`) a y x2 − 4x + 3 0 1 5. x ln xdx. (DS. −0, 25) 0 3 xdx 2√4 6. √ 4 . (DS. 125) x2 − 4 3 2 2 dx 7. . (DS. Phˆn k`) a y (x − 1)2 0 2 xdx 8. . (DS. Phˆn k`) a y x2 − 1 −2 2 x3 dx 16 9. √ . (DS. ’ ˜ ). Chı dˆ n. D˘t x = 2 sin t. a a . 4 − x2 3 0 0 e1/x 2 10. dx. (DS. − ) x3 e −1
  • 116. 11.4. T´ phˆn suy rˆng ıch a o . 115 1 e1/x11. dx. (DS. Phˆn k`) a y x3 0 1 dx12. . (DS. π) x(1 − x) 0 b dx13. ; a < b. (DS. π) (x − a)(b − x) a 1 114. x ln2 xdx. (DS. ) 4 0 Khao s´t su. hˆi tu cua c´c t´ phˆn suy rˆng sau dˆy. ’ a . o . ’ a ıch a . o . a 1 cos2 x15. √ 3 dx. (DS. Hˆi tu) o . . 1 − x2 0 1 √ ln(1 + 3 x16. dx. (DS. Hˆi tu) o . . esin x − 1 0 1 dx17. √ . x (DS. Hˆi tu) o . . e −1 0 1 √ xdx18. . (DS. Hˆi tu) o . . esin x −1 0 1 x2 dx19. . (DS. Phˆn k`) a y 3 (1 − x2 )5 0 1 x3 dx20. . (DS. Phˆn k`) a y 3 (1 − x2 )5 0
  • 117. 116 Chu.o.ng 11. T´ phˆn x´c dinh Riemann ıch a a . 1 dx 21. . (DS. Phˆn k`) a y ex − cos x 0 π/4 ln(sin 2x) 22. √5 dx. (DS. Hˆi tu) o . . x 0 1 ln x 23. √ dx. (DS. Hˆi tu) o . . x 0 Chı dˆ n. Su. dung hˆ th´.c lim xα ln x = 0 ∀ α > 0 ⇒ c´ thˆ lˆy ’ ˜ a ’ . e u . ’ ´ o e a x→0+0 1 |lnx| 1 ’ α = ch˘ng han ⇒ √ < 3/4 . a . 4 x x 1 sin x 24. dx. (DS. Phˆn k`) a y x2 0 2 dx 25. √ . (DS. Hˆi tu) o . . x − x3 0 2 (x − 2) 26. dx. (DS. Phˆn k`) a y x2 − 3x2 + 4 1 1 dx 27. . (DS. Hˆi tu) o . . x(ex − e−x ) 0 2 16 + x4 28. dx. (DS. Hˆi tu) o . . 16 − x4 0 1√ ex − 1 29. dx. (DS. Hˆi tu) o . . sin x 0 1 3 ln(1 + x) 30. dx. (DS. Phˆn k`) a y 1 − cos x 0
  • 118. Chu.o.ng 12 a a ` e ´T´ phˆn h`m nhiˆu biˆn ıch e 12.1 T´ phˆn 2-l´.p . . . . . . . . . . . . . . . . 118 ıch a o 12.1.1 Tru.`.ng ho.p miˆn ch˜. nhˆt . . . . . . . . . 118 o . `e u a . 12.1.2 Tru.`.ng ho.p miˆn cong . . . . . . . . . . . . 118 o . `e 12.1.3 Mˆt v`i u.ng dung trong h` hoc . . . . . . 121 o a ´ . . ınh . 12.2 T´ phˆn 3-l´.p . . . . . . . . . . . . . . . . 133 ıch a o 12.2.1 Tru.`.ng ho.p miˆn h` hˆp . . . . . . . . . 133 o . ` ınh o e . 12.2.2 Tru.`.ng ho.p miˆn cong . . . . . . . . . . . . 134 o . ` e 12.2.3 . . . . . . . . . . . . . . . . . . . . . . . . 136 12.2.4 Nhˆn x´t chung . . . . . . . . . . . . . . . . 136 a e . 12.3 T´ phˆn d u.`.ng . . . . . . . . . . . . . . . 144 ıch a o 12.3.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 144 a . ıa ’ 12.3.2 T´ t´ phˆn du.`.ng . . . . . . . . . . . . 146 ınh ıch a o 12.4 T´ phˆn m˘t . . . . . . . . . . . . . . . . . 158 ıch a a . 12.4.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 158 a . ıa ’ 12.4.2 Phu.o.ng ph´p t´ t´ phˆn m˘t . . . . . . 160 a ınh ıch a a .
  • 119. 118 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 12.4.3 Cˆng th´.c Gauss-Ostrogradski . . . . . . . 162 o u 12.4.4 Cˆng th´.c Stokes . . . . . . . . . . . . . . . 162 o u 12.1 T´ phˆn 2-l´.p ıch a o 12.1.1 Tru.`.ng ho.p miˆn ch˜. nhˆt o . ` e u a . Gia su. ’ ’ D = [a, b] × [c, d] = {(x, y) : a x b, c y d} v` h`m f(x, y) liˆn tuc trong miˆn D. Khi d´ t´ phˆn 2-l´.p cua a a e . ` e o ıch a o ’ h`m f(x, y) theo miˆn ch˜. nhˆt a ` e u a . D = {(x, y) : a x b; c y d} du.o.c t´ theo cˆng th´.c . ınh o u b d f(M )dxdy = dx f (M)dy; (12.1) D a c d b f(M )dxdy = dy f (M)dx, M = (x, y). (12.2) D c a ` Trong (12.1): dˆu tiˆn t´ t´ch phˆn trong I(x) theo y xem x l` h˘ng a e ınh ı a a a` ´ o o ıch a e ´ ’ sˆ, sau d´ t´ phˆn kˆt qua thu du . .o.c I(x) theo x. Dˆi v´.i (12.2) ta ´ o o c˜ng tiˆn h`nh tu.o.ng tu. nhu.ng theo th´. tu. ngu.o.c lai. u ´ e a . . u . . . 12.1.2 Tru.`.ng ho.p miˆn cong o . ` e Gia su. h`m f (x, y) liˆn tuc trong miˆn bi ch˘n ’ ’ a e . ` . a e . D = {(x, y) : a x b; ϕ1(x) y ϕ2 (x)}
  • 120. 12.1. T´ phˆn 2-l´.p ıch a o 119trong d´ y = ϕ1 (x) l` biˆn du.´.i, y = ϕ2(x) l` biˆn trˆn, ho˘c o a e o a e e a . D = {(x, y) : c y d; g1 (y) x g2 (y)}trong d´ x = g1 (y) l` biˆn tr´i c`n x = g2 (y) l` biˆn phai, o. dˆy o a e a o a e ’ ’ a ’ ´ ` ’ta luˆn gia thiˆt c´c h`m ϕ1, ϕ2 , g1 , g2 dˆu liˆn tuc trong c´c khoang o e a a e e . atu.o.ng u.ng. Khi d´ t´ phˆn 2-l´.p theo miˆn D luˆn luˆn tˆn tai. ´ o ıch a o `e o o ` . o ’ Dˆ t´ t´ phˆn 2-l´ e ınh ıch a o.p ta c´ thˆ ´p dung mˆt trong hai phu.o.ng o ea ’ o . .ph´p sau. a 1+ Phu.o.ng ph´p Fubini du.a trˆn dinh l´ Fubini vˆ viˆc du.a t´ a . e . y ` e e . ıch .p vˆ t´ phˆn l˘p. Phu.o.ng ph´p n`y cho ph´p ta du.a t´chphˆn 2-l´ ` ıch a a a o e a a e ı .phˆn 2-l´.p vˆ t´ phˆn l˘p theo hai th´. tu. kh´c nhau: a o ` ıch a a e . u . a b ϕ2 (x) b ϕ2 (x) f (M)dxdy = f(M)dy dx = dx f (M)dy, (12.3) D a ϕ1 (x) a ϕ1 (x) d g2 (y) d g2 (y) f (M)dxdy = f(M )dx dy = dy f (M)dx. (12.4) D c g1 (y) c g1 (y)T`. (12.3) v` (12.4) suy r˘ng cˆn cua c´c t´ch phˆn trong biˆn thiˆn u a ` a a . ’ a ı a e´ e ´v` phu thuˆc v`o biˆn m` khi t´nh t´ phˆn trong, n´ du . a . o a e a ı ıch a o .o.c xem l` a . o’ a ’ ıkhˆng dˆi. Cˆn cua t´ch phˆn ngo`i luˆn luˆn l` h˘ng sˆ. o . a a o o a ` a o´ Nˆu trong cˆng th´.c (12.3) (tu.o.ng u.ng: (12.4)) phˆn biˆn du.´.i ´ e o u ´ `a e o `hay phˆn biˆn trˆn (tu a e e .o.ng u.ng: phˆn biˆn tr´i hay phai) gˆm t`. mˆt ´ ` a e a ’ ` o u o . ´ phˆn v` mˆ i phˆn c´ phu.o.ng tr` riˆng th` miˆn D cˆn chia th`nhsˆ a o ` a o ˜ ` o a ınh e ı e ` ` a anh˜u .ng miˆn con bo.i c´c du.`.ng th˘ng song song v´.i truc Oy (tu.o.ng ` e ’ a o ’ a o .u´.ng: song song v´.i truc Ox) sao cho mˆ i miˆn con d´ c´c phˆn biˆn o ˜ o ` e o a `a e .du.´.i hay trˆn (tu.o.ng u.ng: phˆn biˆn tr´i, phai) dˆu chı du.o.c biˆu o e ´ ` a e a ’ ` e ’ . e’diˆn bo.i mˆt phu.o.ng tr` ˜ e ’ o. ınh. + 2 Phu .o.ng ph´p dˆi biˆn. Ph´p dˆi biˆn trong t´ch phˆn 2-l´.p a o e ’ ´ e o e ’ ´ ı a odu.o.c thu.c hiˆn theo cˆng th´.c . . e . o u D(x, y) f(M )dxdy = f[ϕ(u, v), ψ(u, v)] dudv (12.5) D(u, v) D D∗
  • 121. 120 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e trong d´ D∗ l` miˆn biˆn thiˆn cua toa dˆ cong (u, v) tu.o.ng u.ng o a ` e e´ e ’ . o . ´ a e’ ´ khi c´c diˆm (x, y) biˆn thiˆn trong D: x = ϕ(u, v), y = ψ(u, v); e e (u, v) ∈ D∗ , (x, y) ∈ D; c`n o ∂x ∂x D(x, y) J= = ∂u ∂v = 0 (12.6) D(u, v) ∂y ∂y ∂u ∂v e ’ a a l` Jacobiˆn cua c´c h`m x = ϕ(u, v), y = ψ(u, v). a Toa dˆ cong thu.`.ng d`ng ho.n ca l` toa dˆ cu.c (r, ϕ). Ch´ng . o . o u ’ a . o . . u liˆn hˆ v´.i toa dˆ Dˆcac bo.i c´c hˆ th´.c x = r cos ϕ, y = r sin ϕ, e e o . . o e . ’ a e u . 0 r < +∞, 0 ϕ < 2π. T` u. (12.6) suy ra J = r v` trong toa dˆ a . o . cu .c (12.5) c´ dang o . . f(M )dxdy = f (r cos ϕ, r sin ϕ)rdrdϕ. (12.7) D D∗ K´ hiˆu vˆ phai cua (12.7) l` I(D∗). C´ c´c tru.`.ng ho.p cu thˆ sau y e e . ´ ’ ’ a o a o . . e ’ dˆy. a (i) Nˆu cu.c cua hˆ toa dˆ cu.c n˘m ngo`i D th` ´ e . ’ e . o . ` . . a a ı ϕ2 r2 (ϕ) I(D∗ ) = dϕ f (r cos ϕ, r sin ϕ)rdr. (12.8) ϕ1 r1 (ϕ) (ii) Nˆu cu.c n˘m trong D v` mˆ i tia di ra t`. cu.c c˘t biˆn ∂D ´ . e ` a a o˜ u . a ´ e a o e . ’ khˆng qu´ mˆt diˆm th` o ı 2π r(ϕ) I(D∗ ) = dϕ f (r cos ϕ, r sin ϕ)rdr. (12.9) 0 0 (iii) Nˆu cu.c n˘m trˆn biˆn ∂D cua D th` ´ e . ` a e e ’ ı ϕ2 r(ϕ) ∗ I(D ) = dϕ f (r cos ϕ, r sin ϕ)rdr. (12.10) ϕ1 0
  • 122. 12.1. T´ phˆn 2-l´.p ıch a o 12112.1.3 Mˆt v`i u.ng dung trong h` o a ´ . . ınh hoc . 1+ Diˆn t´ SD cua miˆn ph˘ng D du.o.c t´nh theo cˆng th´.c e ıch . ’ ` e ’ a . ı o u SD = dxdy ⇒ SD = rdrdϕ. (12.11) D D∗ 2+ Thˆ t´ vˆt thˆ h` tru th˘ng d´.ng c´ d´y l` miˆn D (thuˆc ’ e ıch a. ’ e ınh . a ’ u o a a ` e o .m˘t ph˘ a . ’ ng Oxy) v` gi´.i han ph´ trˆn bo.i m˘t z = f (x, y) > 0 du.o.c a a o . ıa e ’ a . .t´ theo cˆng th´ ınh o u .c V = f (x, y)dxdy. (12.12) D 3+ Nˆu m˘t (σ) du.o.c cho bo.i phu.o.ng tr`nh z = f (x, y) th` diˆn ´ e a . . ’ ı ı e . ıch ’ a n´ du.o.c biˆu diˆn bo.i t´ phˆn 2-l´.pt´ cu o . e’ ˜ e ’ ıch a o Sσ = 1 + (fx )2 + (fy )2dxdy, (12.13) D(x,y) ´ o o ’ a . e a . ’trong d´ D(x, y) l` h` chiˆu vuˆng g´c cua m˘t (σ) lˆn m˘t ph˘ng o a ınh e atoa dˆ Oxy. . o . CAC V´ DU ´ I .V´ du 1. T´ t´ phˆn ı . ınh ıch a xydxdy, D = {(x, y) : 1 x 2; 1 y 2}. D Giai. Theo cˆng th´.c (12.2): ’ o u 2 2 xydxdy = dy xydx. D 1 1
  • 123. 122 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e a o o’ T´ t´ phˆn trong (xem y l` khˆng dˆi) ta c´ ınh ıch a o 2 x2 2 1 I(x) = xydx = y = 2y − y. 2 1 2 1 Bˆy gi`. t´ t´ phˆn ngo`i: a o ınh ıch a a 2 1 9 xydxdy = 2y − y dy = · 2 4 D 1 V´ du 2. T´ t´ phˆn ı . ınh ıch a xydxdy nˆu D du.o.c gi´.i han bo.i c´c ´ e . o . ’ a D du.`.ng cong y = x − 4, y 2 = 2x. o Giai. B˘ng c´ch du.ng c´c du.`.ng gi˜.a c´c giao diˆm A(8, 4) v` ’ ` a a . a o u a ’ e a ’ B(2, −2) cua ch´ng, ban doc s˜ thu du . u .o.c miˆn lˆy t´ phˆn D. ` a ıch a e ´ . . e ´ a e ` ´ ´ ´ ´ Nˆu dˆu tiˆn lˆy t´ phˆn theo x v` tiˆp dˆn lˆy t´ phˆn theo e a ıch a a e e a ıch a y th` t´ phˆn theo miˆn D du.o.c biˆu diˆn bo.i mˆt t´ch phˆn bˆi ı ıch a `e . ’ e ˜ e ’ o ı . a o . 4 y4 I= xydxdy = ydy xdx, D −2 y 2 /2 trong d´ doan [−2, 4] l` h`nh chiˆu cua miˆn D lˆn truc Oy. T`. d´ o . a ı ´ e ’ ` e e . u o 4 4 x2 y4 1 y4 I= y dy = y (y + 4)2 − dy = 90. 2 y 2 /2 2 4 −2 −2 Nˆu t´ t´ phˆn theo th´. tu. kh´c: dˆu tiˆn theo y, sau d´ theo ´ e ınh ıch a u . a ` a e o ı ` a `e a ` x th` cˆn chia miˆn D th`nh hai miˆn con bo e ’.i du.`.ng th˘ng qua B v` o ’ a a song song v´o.i truc Oy v` thu du.o.c a . . √ √ 2 2x 8 2x I= + = xdx ydy + xdx ydy √ D1 D2 0 − 2x 2 x−4 2 8 √ y2 2x = xdx · 0 + x dx = 90. 2 x−4 0 2
  • 124. 12.1. T´ phˆn 2-l´.p ıch a o 123 Nhu. vˆy t´ phˆn 2-l´.p d˜ cho khˆng phu thuˆc th´. tu. t´nh t´ch a ıch a . o a o . o . u . ı ı a a ` a o u. tu. t´ch phˆn dˆ khˆng phai chiaphˆn. Do vˆy, cˆn chon mˆt th´ . ı a e o’ ’ . . . `miˆn. eV´ du 3. T´ t´ phˆn ı . ınh ıch a (y − x)dxdy. trong d´ miˆn D du.o.c o ` e . D 1 7gi´.i han bo.i c´c du.`.ng th˘ng y = x + 1, y = x − 3, y = − x + , o . ’ a o ’ a 3 3 1y = − x + 5. 3 Giai. Dˆ tr´nh su. ph´.c tap, ta su. dung ph´p dˆi biˆn u = −y − x; ’ ’ e a . u . ’ . e o e ’ ´ 1v = y + x v` ´p dung cˆng th´.c (12.5). Qua ph´p dˆi biˆn d˜ chon, aa . o u ’ ´ e o e a . 3du.`.ng th˘ng y = x + 1 biˆn th`nh du.`.ng th˘ng u = 1; c`n y = x − 3 o ’ a ´ e a o a’ o ´biˆn th`nh u = −3 trong m˘t ph˘ e a a. ’ ng Ouv; tu.o.ng tu., c´c du.`.ng th˘ng a . a o ’ a 1 7 1 7y = − x + , y = − x + 5 biˆn th`nh c´c du.`.ng th˘ng v = , v = 5. e´ a a o ’ a 3 3 3 3 o `Do d´ miˆn D tro a e ∗ ’. th`nh miˆn D∗ = [−3, 1] × 7 , 5 . Dˆ d`ng thˆy ` e ˜ a e ´ a 3 D(x, y) 3 `r˘ng a = − . Do d´ theo cˆng th´.c (12.5): o o u D(u, v) 4 1 3 3 3 3 (y − x)dxdy = u+ v − − u+ v dudv 4 4 4 4 4 D D∗ 5 4 3 3 = ududv = dv udu = −8. 4 4 D∗ 7/3 −3 Nhˆn x´t. Ph´p dˆi biˆn trong t´ phˆn hai l´.p nh˘m muc d´ch a e . e o e’ ´ ıch a o ` a . ıdo.n gian h´a miˆn lˆy t´ phˆn. C´ thˆ l´c d´ h`m du.´.i dˆu t´ch ’ o ` a ıch a e ´ ’ o e u o a ´ o a ıphˆn tro. nˆn ph´.c tap ho.n. a ’ e u .V´ du 4. T´ t´ phˆn ı . ınh ıch a (x2 + y 2)dxdy, trong d´ D l` h`nh tr`n o a ı o Dgi´.i han bo.i du.`.ng tr`n x2 + y 2 = 2x. o . ’ o o Giai. Ta chuyˆn sang toa dˆ cu.c v` ´p dung cˆng th´.c (12.7). ’ ’ e . o . . a a . o uCˆng th´.c liˆn hˆ (x, y) v´.i toa dˆ cu.c (r, ϕ) v´.i cu.c tai diˆm O(0, 0) o u e e . o . o . . o . . e ’
  • 125. 124 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e c´ dang o . x = r cos ϕ, y = r sin ϕ. (12.14) Thˆ (12.14) v`o phu.o.ng tr` du.`.ng tr`n ta thu du.o.c r2 = 2r cos ϕ ⇒ e´ a ınh o o . r = 0 ho˘c r = 2 cos ϕ (dˆy l` phu a a a .o.ng tr` du.`.ng tr`n trong toa dˆ ınh o o . . o . cu .c). Khi d´ o . π π D∗ = (r, ϕ) : − ϕ ,0 r 2 cos ϕ 2 2 T`. d´ thu du.o.c u o . π/2 2 cos ϕ 2 2 3 I= (x + y )dxdy = r drdϕ = dϕ r3 dr D D∗ −π/2 0 π/2 π/2 r4 2 cos ϕ 3π = dϕ = 4 cos4 ϕf ϕ = · 4 0 2 −π/2 −π/2 Nhˆn x´t. Nˆu lˆy cu.c tai tˆm h` tr`n th` a e . ´ ´ e a . . a ınh o ı x − 1 = r cos ϕ y = r sin ϕ D∗ = (r, ϕ) : 0 r 1, 0 ϕ 2π} v` do x2 + y 2 = 1 + 2r cos ϕ + r2 nˆn a e I= r(1 + 2r cos ϕ + r2 )drdϕ D∗ 2π 1 3π = dϕ (r + 2r2 cos ϕ + r3 )dr = · 2 0 0 V´ du 5. T´ thˆ t´ vˆt thˆ T gi´.i han bo.i paraboloid z = x2 + y 2, ı . ’ ınh e ıch a . ’ e o . ’ a . . a a a . ’ m˘t tru y = x2 v` c´c m˘t ph˘ng y = 1, z = 0. a
  • 126. 12.1. T´ phˆn 2-l´.p ıch a o 125 ’ ´ e ’ a . ’ e e a . ’ Giai. H` chiˆu cua vˆt thˆ T lˆn m˘t ph˘ng Oxy l` ınh a a D(x, y) = (x, y) : −1 x 1, x2 y 1 .Do d´ ´p dung (12.12) ta c´ oa . o 1 1 2 2 V (T ) = zdxdy = (x + y )dxdy = dx (x2 + y 2)dy D(x,y) D(x,y) −1 x2 1 y3 1 88 = x2 y + dx = · 3 x2 105 −1V´ du 6. T` diˆn t´ m˘t cˆu b´n k´nh R v´.i tˆm tai gˆc toa dˆ. ı . ım e ıch a ` . . a a ı o a ´ . o . o . ’ i. Phu.o.ng tr` m˘t cˆu d˜ cho c´ dang Gia ınh a a . ` a o . x2 + y 2 + z 2 = R2 .Do d´ phu.o.ng tr` nu.a trˆn m˘t cˆu l` o ınh ’ e a ` a . a z= R2 − x2 − y 2.Do t´ dˆi x´.ng nˆn ta chı t´ diˆn t´ nu.a trˆn l` du. Ta c´ ´ ınh o u e ’ ınh e ıch ’ . e a ’ o Rdxdy ds = 1 + zx2 + zy 2 dxdy = · R2 − x2 − y 2 Miˆn lˆy t´ phˆn D(x, y) = {(x, y) : x2 + y 2 ` a ıch a e ´ R2 }. Do d´ o x = r cos ϕ R S=2 dxdy = y = r sin ϕ R2 − x2 − y 2 D(x,y) J =r 2π R rdr = 2R dϕ √ R2 − r2 0 0 √ R = 4πR − R2 − r2 0 = 4πR2 .
  • 127. 126 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e V´ du 7. T´ diˆn t´ phˆn m˘t tru x2 = 2z gi´.i han bo.i giao ı . ınh e ıch ` . a a . . o . ’ ´ e ’ tuyˆn cua m˘t tru d´ v´ a a .i c´c m˘t ph˘ng x − 2y = 0, y = 2x, x = 2√2. ’ . . o o a . a ’ ˜ a ` e ´ a ´ e ’ ` Giai. Dˆ thˆy r˘ng h`nh chiˆu cua phˆn m˘t d˜ nˆu l` tam gi´c ı a a a e a . a .i c´c canh n˘m trˆn giao tuyˆn cua m˘t ph˘ng Oxy v´.i c´c m˘t v´ a . o ` a e ´ ’ e a ’ a o a a . . ph˘’ ng d˜ cho. a a x2 T`. phu.o.ng tr` m˘t tru ta c´ z = , do vˆy u ınh a . . o a . 2 ∂z ∂z √ = x, = 0 → dS = 1 + x2dxdy. ∂x ∂y T`. d´ suy r˘ng u o ` a √ √ 2 2 2x 2 2 √ 3 √ S= 1 + x2dx dy = x 1 + x2dx = 13. 2 0 x/2 0 ` ˆ BAI TAP . T` cˆn cua t´ phˆn hai l´.p ım a . ’ ıch a o f (x, y)dxdy theo miˆn D gi´.i ` e o D han bo.i c´c du.`.ng d˜ chı ra . (Dˆ ng˘n gon ta k´ hiˆu f(x, y) = f (−)). . ’ a o a ’ ’ a e ´ . y e . 1. x = 3, x = 5, 3x − 2y + 4 = 0, 3x − 2y + 1 = 0. 3x+4 5 5 (DS. dx f (−)dy) 3 3x+1 5 2. x = 0, y = 0, x + y = 2 2 2−x (DS. dx f (−)dy) 0 0
  • 128. 12.1. T´ phˆn 2-l´.p ıch a o 1273. x2 + y 2 1, x 0, y 0. √ 1 1−x2 (DS. dx f (−)dy) 0 04. x + y 1, x − y 1, x 0. 1 1−x (DS. dx f (−)dy) 0 x−15. y x2, y 4 − x2 . √ 2 4−x2 (DS. dx f (−)dy) √ − 2 x2 x2 y 26. + 1. 4 9 3 √ +2 2 4−x2 (DS. dx f (−)dy) √ −2 −3 4−x2 2 √7. y = x2, y = x. √ 1 x (DS. dx f (−)dy) 0 x28. y = x, y = 2x, x + y = 6. 2 2x 3 6−x (DS. dx f(−)dy + dx f (−)dy) 0 x 2 x Thay dˆi th´. tu. t´ phˆn trong c´c t´ phˆn ’ o u . ıch a a ıch a
  • 129. 128 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 4 4 4 x 9. dy f(−)dx. (DS. dx f (−)dy) 0 y 2 2 √ 0 1−x2 1 y−1 10. dx f(−)dy. (DS. dy f (−)dx) −1 x+1 0 √ − 1−y 2 √ 1 2−x2 1 y 2 2−y 11. dx f(−)dy. (DS. dy f dx + dy f dx) 0 x 0 0 1 0 2 y 1 2 2 2 12. dy fdx. (DS. dx f dy + dx f dy) 1 1/y 1/2 1/x 1 x T´ c´c t´ phˆn l˘p sau ınh a ıch a a . 1 2x 1 13. dx (x − y + 1)dy. (DS. ) 3 0 x 4 y y3 14. dy dx. (DS. 6π) x2 + y 2 −2 0 0 y2 15. dy (x + 2y)dx. (DS. −11, 2) 2 0 5 5−x 506 16. dx 4 + x + ydy. (DS. ) 15 0 0 4 2 dy 25 17. dx . (DS. ) (x + y)2 24 3 1 √ a 2 ax 344 4 18. dx (x2 + y 2)dy. (DS. a) √ 105 0 −2 ax
  • 130. 12.1. T´ phˆn 2-l´.p ıch a o 129 2π a πa219. dϕ rdr. (DS. ) 2 0 a sin ϕ √ 1 1−x2 π20∗. dx 1 − x2 − y 2dy. (DS. ) 6 0 0 T´ c´c t´ phˆn 2-l´.p theo c´c h` ch˜. nhˆt d˜ chı ra. ınh a ıch a o a ınh u a a ’. 521. (x + y 2)dxdy; 2 x 3, 1 y 2. (DS. 4 ) 6 D 522. (x2 + y)dxdy; 1 x 2, 0 y 1. (DS. 2 ) 6 D 223. (x2 + y 2 )dxdy; 0 x 1, 0 y 1. (DS. ) 3 D 3y 2 dxdy π24. ;0 x 1, 0 y 1. (DS. ) 1 + x2 4 D π π25. sin(x + y)dxdy; 0 x ,0 y . (DS. 2) 2 2 D 126. xexy dxdy; 0 x 1, −1 y 0. (DS. ) e D dxdy 427. ;1 x 2, 3 y 4. (DS. ln ) (x − y)2 3 D T´ c´c t´ phˆn 2-l´.p theo miˆn D gi´.i han c´c du.`.ng d˜ chı ınh a ıch a o ` e o . a o a ’ra 128. xydxdy; y = 0, y = x, x = 1. (DS. ) 8 D 129. xydxdy; y = x2 , x = y 2. (DS. ) 12 D
  • 131. 130 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 7 30. xdxdy; y = x3, x + y = 2, x = 0. (DS. ) 15 D 5 31. xdxdy; xy = 6, x + y − 7 = 0. (DS. 20 ) 6 D 3 32. y 2xdxdy; x2 + y 2 = 4, x + y − 2 = 0. (DS. 1 ) 5 D 5π 33. (x + y)dxdy; 0 y π, 0 x sin y. (DS. ) 4 D π 1 34. sin(x + y)dxdy; x = y, x + y = , y = 0. (DS. ) 2 2 D e−y dxdy; D l` tam gi´c v´.i dınh O(0, 0), B(0, 1), A(1, 1). 2 35. a a o ’ D 1 1 (DS. − + ) 2e 2 36. xydxdy; D l` h`nh elip 4x2 + y 2 a ı 4. (DS. 0) D √ 4a5 37. x2ydxdy; y = 0, y = 2ax − x2 . (DS. ) 5 D xdxdy π 1 38. ; y = x, x = 2, x = 2y. (DS. − 2arctg ) x2 + y 2 2 2 D √ 2 39. x + ydxdy; x = 0, y = 0, x + y = 1. (DS. ) 5 D 4 40. (x − y)dxdy; y = 2 − x2, y = 2x − 1. (DS. 4 ) 15 D 1 41. (x + 2y)dxdy; y = x, y = 2x, x = 2, x = 3. (DS. 25 ) 3 D
  • 132. 12.1. T´ phˆn 2-l´.p ıch a o 131 9π42. xdxdy; x = 2 + sin y, x = 0, y = 0, y = 2π. (DS. ) 2 D 443. xydxdy; (x − 2)2 + y 2 = 1. (DS. ) 3 D dxdy44. √ a ınh o a ı ` ; D l` h` tr`n b´n k´nh a n˘m trong g´c vuˆng I a o o 2a − x D 8 √v` tiˆp x´c v´.i c´c truc toa dˆ. (DS. a 2a) ´ a e u o a . . o . 345. ydxdy; x = R(t − sin t), y = R(1 − cos t), 0 t a ` 2π (l` miˆn e D 5gi´.i han bo.i v`m cua xicloid.) (DS. πR3 ) o . ’ o ’ 2 2πR y=f (x) ’ a˜ Chı dˆ n. ydxdy = dx ydy D 0 0 Chuyˆn sang toa dˆ cu.c v` t´nh t´ch phˆn trong toa dˆ m´.i ’ e . o . a ı . ı a . o o . πR446. (x2 + y 2 )dxdy; D : x2 + y 2 R2 , y 0. (DS. ) 4 D 2 +y 2 π47. ex dxdy; D : x2 + y 2 1, x 0, y 0. (DS. (e − 1)) 4 D 2 +y 2 248. ex dxdy; D : x2 + y 2 R2 . (DS. 2π(eR − 1)) D 1 449. 1 − x2 − y 2dxdy; D : x2 + y 2 x. (DS. π− ) 4 3 D 1 − x2 − y 250. dxdy; D : x2 + y 2 1, x 0, y 0. 1 + x2 + y 2 D π(π − 2) (DS. ) 2
  • 133. 132 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e ln(x2 + y 2) 51. dxdy; D : 1 x2 + y 2 e. (DS. 2π) x2 + y 2 D 52. (x2 + y 2)dxdy; D gi´.i han bo.i c´c du.`.ng tr`n o . ’ a o o D 5π x2 + y 2 + 2x − 1 = 0, x2 + y 2 + 2x = 0. (DS. ) 2 ’ ˜ Chı dˆ n. D˘t x − 1 = r cos ϕ, y = r sin ϕ. a a . T´ thˆ t´ cua vˆt thˆ gi´.i han bo.i c´c m˘t d˜ chı ra. ’ ınh e ıch ’ a . ’ e o . ’ a a a ’ . 1 53. x = 0, y = 0, z = 0, x + y + z = 1. (DS. ) 6 1 54. x = 0, y = 0, z = 0, x + y = 1, z = x2 + y 2 . (DS. ) 6 88 55. z = x2 + y 2, y = x2, y = 1, z = 0. (DS. ) 105 2 56. z = x2 + y 2 , x2 + y 2 = a2, z = 0. (DS. πa3) 3 πa4 57. z = x2 + y 2, x2 + y 2 = a2, z = 0. (DS. ) 2 4a3 58. z = x, x2 + y 2 = a2, z = 0. (DS. ) 3 1 59. z = 4 − x2 − y 2, x = ±1, y = ±1. (DS. 13 ) 3 11 60. 2 − x − y − 2z = 0, y = x2 , y = x. (DS. ) 120 61. x2 + y 2 = 4x, z = x, z = 2x. (DS. 4π) ınh e ıch a . ` a a a ’ T´ diˆn t´ c´c phˆn m˘t d˜ chı ra. . ` a . ’ a ` a o ` a 62. Phˆn m˘t ph˘ng 6x + 3y + 2z = 12 n˘m trong g´c phˆn t´m I. a a (DS. 14) ` a a ’ 63. Phˆn m˘t ph˘ng x + y + z = 2a n˘m trong m˘t tru x2 + y 2 = a2. . √ a ` a a . . 2 (DS. 2a 3)
  • 134. 12.2. T´ phˆn 3-l´.p ıch a o 13364. Phˆn m˘t paraboloid z = x2 + y 2 n˘m trong m˘t tru x2 + y 2 = 4. ` a a. ` a a . . π √ (DS. (17 17 − 1)) 665. Phˆn m˘t 2z = x2 + y 2 n˘m trong m˘t tru x2 + y 2 = 1. ` a a. ` a a . . 2 √ (DS. (2 2 − 1)π) 366. Phˆn m˘t n´n z = x2 + y 2 n˘m trong m˘t tru x2 + y 2 = a2 . ` a a o . √ ` a a . . 2 (DS. πa 2)67. Phˆn m˘t cˆu x2 + y 2 + z 2 = R2 n˘m trong m˘t tru x2 + y 2 = Rx. ` a a ` . a ` a a . . 2 (DS. 2R (π − 2))68. Phˆn m˘t n´n z 2 = x2 + y 2 n˘m trong m˘t tru x2 + y 2 = 2x. `a a o . ` a a. . √ (DS. 2 2π)69. Phˆn m˘t tru z 2 = 4x n˘m trong g´c phˆn t´m th´ I v` gi´.i han `a a . . ` a o ` a a u a o . 4 √bo.i m˘t tru y 2 = 4x v` m˘t ph˘ng x = 1. (DS. (2 2 − 1)) ’ a. . a a. ’ a 370. Phˆn m˘t cˆu x2 + y 2 + z 2 = R2 n˘m trong m˘t tru x2 + y 2 = a2 ` a a ` . a ` a a . . √(a R). (DS. 4πa(a − a 2 − R2 ))12.2 T´ phˆn 3-l´.p ıch a o12.2.1 Tru.`.ng ho.p miˆn h` hˆp o . ` e ınh o .Gia su. miˆn D ⊂ R3: ’ ’ ` eD = [a, b] × [c, d] × [e, g] = {(x, y, z) : a x b, c y d, e z g}v` h`m f (x, y, z) liˆn tuc trong D. Khi d´ t´ phˆn 3-l´.p cua h`m a a e . o ıch a o ’ a ` e .o.c t´ theo cˆng th´.cf(x, y, z) theo miˆn D du . ınh o u b d g f(x, y, z)dxdydz = f (x, y, z)dz dy dx D a c e b d g = dx dy f (M)dx. (12.15) a c e
  • 135. 134 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e T`. (12.15) suy ra c´c giai doan t´ t´ch phˆn 3-l´.p: u a . ınh ı a o g ` (i) Dˆu tiˆn t´ I(x, y) = a e ınh f (M)dz; e d ´ (ii) Tiˆp theo t´ I(x) = e ınh I(x, y)dy; c b (iii) Sau c`ng t´ t´ch phˆn I = u ınh ı a I(x)dx. a Nˆu t´ phˆn (12.15) du.o.c t´nh theo th´. tu. kh´c th` c´c giai doa n ´ e ıch a . ı u . a ı a . t´ vˆ n tu.o.ng tu.: dˆu tiˆn t´nh t´ch phˆn trong, tiˆp dˆn t´ t´ ınh a˜ . a ` e ı ı a ´ e ınh ıch e ´ a u.a v` sau c`ng l` t´ t´ch phˆn ngo`i. phˆn gi˜ a u a ınh ı a a 12.2.2 Tru.`.ng ho.p miˆn cong o . ` e 1+ Gia su. h`m f(M ) liˆn tuc trong miˆn bi ch˘n ’ ’ a e . ` . a e . D = (x, y, z) : a x b, ϕ1(x) y ϕ2 (x), g1 (x, y) z g2 (x, y) . Khi d´ t´ phˆn 3-l´.p cua h`m f(M) theo miˆn D du.o.c t´ theo o ıch a o ’ a ` e . ınh cˆng th´.c o u b ϕ2 (x) g2 (x,y) f(M )dxdydz = f (M)dx dy dx (12.16) D a ϕ1 (x) g1 (x,y) ho˘c a . g2 (x,y) f(M )dxdydz = dxdy f (M)dz, (12.17) D D(x,y) g1 (x,y) e´ o o ’ e a. ’ trong d´ D(x, y) l` h` chiˆu vuˆng g´c cua D lˆn m˘t ph˘ng Oxy. o a ınh a Viˆc t´ t´ phˆn 3-l´.p du.o.c quy vˆ t´nh liˆn tiˆp ba t´ch phˆn thˆng e ınh ıch a . o . ` ı e e e ´ ı a o
  • 136. 12.2. T´ phˆn 3-l´.p ıch a o 135thu.`.ng theo (12.16) t`. t´ phˆn trong, tiˆp dˆn t´ch phˆn gi˜.a v` o u ıch a ´ ´ e e ı a u asau c`ng l` t´ t´ phˆn ngo`i. Khi t´nh t´ phˆn 3-l´ u a ınh ıch a a ı ıch a o.p theo cˆng oth´.c (12.17): dˆu tiˆn t´ t´ phˆn trong v` sau d´ c´ thˆ t´nh t´ch u ` a e ınh ıch a a o o e ı ’ ıphˆn 2-l´ a o.p theo miˆn D(x, y) theo c´c phu.o.ng ph´p d˜ c´ trong 12.1. ` e a a a o 2 Phu.o.ng ph´p dˆi biˆn. Ph´p dˆi biˆn trong t´ch phˆn 3-l´.p + a o e ’ ´ e o e ’ ´ ı a o .o.c tiˆn h`nh theo cˆng th´.cdu . ´ e a o u f(M )dxdydz = f ϕ(u, v, w), ψ(u, v, w), χ(u, v, w) × D D∗ D(x, y, z) × dudvdw, (12.18) D(u, v, w)trong d´ D∗ l` miˆn biˆn thiˆn cua toa dˆ cong u, v, w tu.o.ng u.ng khi o a ` e ´ e e ’ . o . ´ a e’ ´c´c diˆm (x, y, z) biˆn thiˆn trong D: x = ϕ(u, v, w), y = ψ(u, v, w), e e D(x, y, z)z = χ(u, v, w), a e ’ a a l` Jacobiˆn cua c´c h`m ϕ, ψ, χ D(u, v, w) ∂ϕ ∂ϕ ∂ϕ ∂u ∂v ∂w D(x, y, z) ∂ψ ∂ψ ∂ψ J= = = 0. (12.19) D(u, v, w) ∂u ∂v ∂w ∂χ ∂χ ∂χ ∂u ∂v ∂w Tru.`.ng ho.p d˘c biˆt cua toa dˆ cong l` toa dˆ tru v` toa dˆ cˆu. o . a. e ’ . o . . a . o . a . o ` . . a (i) Bu.´.c chuyˆn t`. toa dˆ Dˆc´c sang toa dˆ tru (r, ϕ, z) du.o.c thu.c o ’ e u . o e a. . o . . . .hiˆn theo c´c hˆ th´ e a e u .c x = r cos ϕ, y = r sin ϕ, z = z; 0 r < +∞, . .0 ϕ < 2π, −∞ < z < +∞. T` u. (12.19) suy ra J = r v` trong toa a .dˆ tru ta c´ o . . o f(M )dxdydz = f r cos ϕ, r sin ϕ, z rdrdϕdz, (12.20) D D∗trong d´ D∗ l` miˆn biˆn thiˆn cua toa dˆ tru tu.o.ng u.ng khi diˆm o a ` e e´ e ’ . o . . ´ ’ e ´(x, y, z) biˆn thiˆn trong D. e e
  • 137. 136 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e (ii) Bu.´.c chuyˆn t`. toa dˆ Dˆc´c sang toa dˆ cˆu (r, ϕ, θ) du.o.c o ’ e u . o e a . . o ` . a . thu .c hiˆn theo c´c hˆ th´.c x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = e a e u . . . r cos θ, 0 r < +∞, 0 ϕ < 2π, 0 θ π. T`. (12.19) ta c´ u o 2 a . o ` J = r sin θ v` trong toa dˆ cˆu ta c´ . a o f(M )dxdydz = D = f r sin θ cos ϕ, r sin θ sin ϕ, r cos θ r2 sin θdrdϕdθ, (12.21) D∗ trong d´ D∗ l` miˆn biˆn thiˆn cua toa dˆ cˆu tu.o.ng u.ng khi diˆm o a ` e e´ e ’ . o ` . a ´ ’ e ´ (x, y, z) biˆn thiˆn trong D. e e 12.2.3 Thˆ t´ cua vˆt thˆ cho´n hˆt miˆn D ⊂ R3 du.o.c t´ theo cˆng ’ e ıch ’ a . ’ e a ´ e ` e . ınh o th´ u.c VD = dxdydz. (12.22) D 12.2.4 Nhˆn x´t chung a . e B˘ng c´ch thay dˆi th´. tu. t´nh t´ch phˆn trong t´ch phˆn 3-l´.p ta s˜ ` a a ’ o u . ı ı a ı a o e .o.c c´c cˆng th´.c tu.o.ng tu. nhu. cˆng th´.c (12.16) dˆ t´nh t´ch thu du . a o u o u ’ e ı ı . phˆn. Viˆc t` cˆn cho t´ch phˆn do.n thˆng thu.`.ng khi chuyˆn t´ a e ım a . . ı a o o ’ e ıch phˆn 3-l´.p vˆ t´ phˆn l˘p du.o.c thu.c hiˆn nhu. dˆi v´.i tru.`.ng ho.p a o ` ıch a a e . . . e . ´ o o o . t´ phˆn 2-l´ ıch a o.p. CAC V´ DU ´ I . V´ du 1. T´ t´ phˆn l˘p ı . ınh ıch a a. 1 1 2 I= dx dy (4 + z)dx. −1 x2 0
  • 138. 12.2. T´ phˆn 3-l´.p ıch a o 137 Giai. Ta t´ liˆn tiˆp ba t´ phˆn x´c dinh thˆng thu.`.ng b˘t ’ ınh e ´ e ıch a a . o o ´ a ` u . t´ phˆn trongdˆu t` ıch a a 2 2 z2 2 I(x, y) = (4 + z)dz = 4z 0 + = 10; 2 0 0 1 1 I(x) = I(x, y)dy = 10 dy = 10(1 − x2); x2 x2 1 1 40 I= I(x)dx = 10(1 − x2 )dx = · 3 −1 −1V´ du 2. T´ t´ phˆn ı . ınh ıch a I= (x + y + z)dxdydz, Dtrong d´ miˆn D du.o.c gi´.i han bo.i c´c m˘t ph˘ng toa dˆ v` m˘t o ` e . o . ’ a a . ’ a . o a a . .ph˘’ ng x + y + z = 1. a Giai. Miˆn D d˜ cho l` mˆt t´. diˆn c´ h`nh chiˆu vuˆng g´c trˆn ’ `e a a o u e o ı . . ´ e o o em˘t ph˘ a . ’ ng Oxy l` tam gi´c gi´.i han bo.i c´c du.`.ng th˘ng x = 0, a a a o . ’ a o ’ a a ´y = 0, x + y = 1. R˜ r`ng l` x biˆn thiˆn t` o a e e u . 0 dˆn 1 (doan [0, 1] l` e´ a . ınh ´ e ’ e . ´h` chiˆu cua D lˆn truc Ox). Khi cˆ dinh x, 0 x 1 th` y biˆn o . ı ´ ethiˆn t`. 0 dˆn 1 − x. Nˆu cˆ dinh ca x v` y (0 x 1, 0 y 1 − x) e u ´ e ´ ´ e o . ’ a ı e ’ ´th` diˆm (x, y, z) biˆn thiˆn theo du o e e .`.ng th˘ng d´.ng t`. m˘t ph˘ng ’ a u u a ’ a .z = 0 dˆn m˘t ph˘ng x + y + z = 1, t´.c l` z biˆn thiˆn t`. 0 dˆn ´ e a . ’ a u a ´ e e u ´ e1 − x − y. Theo cˆng th´.c (12.16) ta c´ o u o 1 1−x 1−x−y I= dx dy (x + y + z)dz. 0 0 0
  • 139. 138 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e ˜ a a ` ´ a Dˆ d`ng thˆy r˘ng e 1 1−x z2 1−x−y I= dx xz + yz + dy 2 0 0 0 1 1 y3 1−x = y − yx2 − xy 2 − dx 2 3 0 0 1 1 1 = (2 − 3x + x3)dx = · 6 8 0 dxdydz V´ du 3. T´ I = ı . ınh , trong d´ miˆn D du.o.c gi´.i o ` e . o (x + y + z)3 D han bo.i c´c m˘t ph˘ng x + z = 3, y = 2, x = 0, y = 0, z = 0. . ’ a a. a’ ’ ` Giai. Miˆn D d˜ cho l` mˆt h`nh l˘ng tru c´ h`nh chiˆu vuˆng e a a o ı . a . o ı ´ e o a ’ g´c lˆn m˘t ph˘ng Oxy l` h` ch˜ o e a a ınh u . nhˆt D(x, y) = (x, y) : 0 a . . x 3, 0 y 2 . V´ o.i diˆm M(x, y) cˆ dinh thuˆc D(x, y) diˆm ’ e ´ o . o ’ e . (x, y, z) ∈ D biˆn thiˆn trˆn du.`.ng th˘ng d´.ng t`. m˘t ph˘ng Oxy ´ e e e o ’ a u u a . a’ ´ e a ’ (z = 0) dˆn m˘t ph˘ng x + z = 3, t´ a a u.c l` z biˆn thiˆn t`. 0 dˆn 3 − x: ´ e e u ´ e . 0 z 3 − x. T`. d´ theo (12.17) ta c´ u o o z=3−x f(M )dxdydz = dxdy (x + y + z + 1)−3 dz D D(x,y) z=0 (x + y + z + 1)−2 3−x 4 ln 2 − 1 = dxdy = · · · = · −2 0 8 D(x,y) V´ du 4. T´ t´ phˆn ı . ınh ıch a (x2 + y 2 + z 2 )dxdydz, trong d´ miˆn o ` e D D du.o.c gi´.i han bo.i m˘t 3(x2 + y 2) + z 2 = 3a2 . . o . ’ a . ’ Giai. Phu .o.ng tr` m˘t biˆn cua D c´ thˆ viˆt du.´.i dang ınh a e ’ ’ ´ o e e o . . x2 y 2 z2 + 2 + √ = 1. a2 b (a 3)2
  • 140. 12.2. T´ phˆn 3-l´.p ıch a o 139D´ l` m˘t elipxoid tr`n xoay, t´.c l` D l` h` elipxoid tr`n xoay. o a a . o u a a ınh o ´ o ’ e a . ’H` chiˆu vuˆng g´c D(x, y) cua D lˆn m˘t ph˘ng Oxy l` h`nh tr`n ınh e o a a ı ox2 + y 2 a2. Do d´ ´p dung c´ch lˆp luˆn nhu. trong c´c v´ du 2 o a . a a . a . a ı . ´ ` ’v` 3 ta thˆy r˘ng khi diˆm M(x, y) ∈ D(x, y) du . o . a a a e .o.c cˆ dinh th` diˆm ´ ı e ’ ’ ` e ´(x, y, z) cua miˆn D biˆn thiˆn trˆn du o e e e .`.ng th˘ng d´.ng M(x, y) t`. ’ a u um˘t biˆn du.´.i cua D a. e o ’ z = − 3(a2 − x2 − y 2) ´dˆn m˘t biˆn trˆn e a . e e z=+ 3(a2 − x2 − y 2).T`. d´ theo (12.17) ta c´ u o o √ + 3(a2 −x2 −y 2 ) I= dxdy (x2 + y 2 + z 2 )dz D(x,y) √ − 3(a2 −x2 −y 2 ) √ = 2a2 3 a2 − x2 − y 2dxdy = |chuyˆn sang toa dˆ cu.c| e’ . o . . x2 +y 2 a2 2π a 2 √ √ √ = 2a 3 a2 − r2 rdrdϕ = a2 3 dϕ (a2 − r2 )1/2rdr r a 0 0 5 4πa = √ · 3V´ du 5. T´ thˆ t´ cua vˆt thˆ gi´.i han bo.i c´c m˘t ph˘ng ı . ’ ınh e ıch ’ a . ’ e o . ’ a a . ’ ax + y + z = 4, x = 3, y = 2, x = 0, y = 0, z = 0. ’ ` Giai. Miˆn D d˜ cho l` mˆt h` luc diˆn trong khˆng gian. N´ e a a o ınh . . e . o o o ınh ´ e o o e a . ’c´ h` chiˆu vuˆng g´c D(x, y) lˆn m˘t ph˘ng Oxy l` h`nh thang a a ıvuˆng gi´.i han bo.i c´c du.`.ng th˘ng x = 0, y = 0, x = 3, y = 2 v` o o . ’ a o ’ a a
  • 141. 140 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e x + y = 4. Do d´ ´p dung (12.17) ta c´ oa . o 4−x−y VD = dxdydz = dxdy dz = (4 − x − y)dxdy D D(x,y) 0 D(x,y) 1 3 2 4−y = dy (4 − x − y)dx + dy (4 − x − y)dx 0 0 1 0 1 2 x2 3 x2 4−y = (4 − y)x − dy + (4 − y)x − dy 2 0 2 0 0 1 1 2 15 1 55 = − 3y dy + (4 − y)2dy = · 2 2 6 0 1 V´ du 6. T´ t´ phˆn ı . ınh ıch a I= z x2 + y 2dxdydz, D trong d´ miˆn D gi´.i han bo.i m˘t ph˘ng y = 0, z = 0, z = a v` m˘t o ` e o . ’ a . ’ a a a . 2 2 tru x + y = 2x (x 0, y 0, a > 0). . Giai. Chuyˆn sang toa dˆ tru ta thˆy phu.o.ng tr` m˘t tru x2 + ’ ’ e . o .. ´ a ınh a . . 2 π y = 2x trong toa dˆ tru c´ dang r = 2 cos ϕ, 0 ϕ . o . o . . (h˜y v˜ h`nh a e ı .c (12.20) ta c´ 2 !). Do d´ theo cˆng th´ o o u o π/2 2 cos ϕ a π/2 2 cos ϕ 2 a2 I= dϕ r dr zdz = dϕ r2 dr 2 0 0 0 0 0 π/2 4a2 8 = cos3 ϕdϕ = a2. 3 9 0 V´ du 7. T´ t´ phˆn ı . ınh ıch a I= (x2 + y 2 )dxdydz, D
  • 142. 12.2. T´ phˆn 3-l´.p ıch a o 141nˆu miˆn D l` nu.a trˆn cua h` cˆu x2 + y 2 + z 2 R2 , z 0. ´ e `e a ’ e ’ ınh ` a ’ e’ Giai. Chuyˆn sang toa dˆ cˆu, miˆn biˆn thiˆn D∗ cua c´c toa dˆ . o ` . a ` e ´ e e ’ a . o. `cˆu tu a .o.ng u.ng khi diˆm (x, y, z) biˆn thiˆn trong D l` c´ dang ´ e’ ´ e e a o . π D∗ : 0 ϕ < 2π, 0 θ , 0 r R. 2T`. d´ u o 2π π/2 R 2 2 2 3 I= r sin θ · r sin θdrdϕdθ = dϕ sin θdθ r4 dr D∗ 0 0 0 4 = πR5 . 15 ` ˆ BAI TAP . T´ c´c t´ phˆn l˘p sau ınh a ıch a a . √ 1 x 2−2x 11. dx ydy dz. (DS. ) 12 0 0 1−x a h a−y a3h2. ydy dx dz. (DS. ) 6 0 0 0 2 2 33. dy xdx z 2dz. (DS. 30) 0 √ 0 2y−y 2 1 1−x 1−x−y dz ln 2 54. dx dy . (DS. − ) (1 + x + y + z)3 2 16 0 0 0 c b a abc 25. dz dy (x2 + y 2 + z 2 )dx. (DS. (a + b2 + c2 ) ) 3 0 0 0
  • 143. 142 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e a a−x a−x−y a5 6. dx dy (x2 + y 2 + z 2)dz. (DS. ) 20 0 0 0 T´ c´c t´ phˆn 3-l´.p theo miˆn D gi´.i han bo.i c´c m˘t d˜ chı ınh a ıch a o ` e o . ’ a a a ’ . ra. 7. (x + y − z)dxdydz; x = −1, x = 1; y = 0, y = 1; D z = 0, z = 2. (DS. −2) 1 8. xydxdydz; x = 1, x = 2; y = −2, y = −1; z = 0, z = . 2 D 8 (DS. − ) 9 dxdydz 9. ; x = 1, x = 2; y = 1, y = 2; z = 1, z = 2. (x + y + z)2 D 1 128 (DS. ln ) 2 125 10. (x + 2y + 3z + 4)dxdydz; x = 0, x = 3; y = 0, y = 2; D z = 0, z = 1. (DS. 54) 1 11. zdxdydz; x = 0, y = 0, z = 0; x + y + z = 1. (DS. ) 24 D 1 12. xdxdydz; x = 0. y = 0, z = 0, y = 1; x + z = 1. (DS. ) 6 D 13. yzdxdydz; x2 + y 2 + z 2 = 1, z 0. (DS. 0) D 14. xydxdydz; x2 + y 2 = 1, z = 0, z = 1 (x 0, y 0). D 1 (DS. ) 8 15. xyzdxdydz; x = 0, y = 0, z = 0, x2 + y 2 + z 2 = 1 D
  • 144. 12.2. T´ phˆn 3-l´.p ıch a o 143 1 (x 0, y 0, z 0). (DS. ) 4816. x2 + y 2dxdydz; x2 + y 2 = z 2 , z = 0, z = 1. (DS. π/6) D17. (x2 + y 2 + z 2 )dxdydz; x = 0, x = a, y = 0, y = b, D abc 2 z = 0, z = c. (DS. (a + b2 + c2 )) 3 √ πh418. ydxdydz; y = x2 + z 2 , y = h, h > 0. (DS. ) 4 D T´ c´c t´ phˆn 3-l´.p sau b˘ng phu.o.ng ph´p dˆi biˆn. ınh a ıch a o ` a a o e’ ´ 4πR519. (x2 + y 2 + z 2 )dxdydz; x2 + y 2 + z 2 R2 . (DS. ) 5 D π20. (x2 + y 2)dxdydz; z = x2 + y 2, z = 1. (DS. ) 6 D21. x2 + y 2 + z 2 dxdydz; x2 + y 2 + z 2 R2 . (DS. πR4 ) D22. z x2 + y 2dxdydz; x2 + y 2 = 2x, y = 0, z = 0, z = 3. D (DS. 8)23. zdxdydz; x2 + y 2 + z 2 R2 , x 0, y 0, z 0. D πR4 (DS. ) 16 16π24. (x2 − y 2 )dxdydz; x2 + y 2 = 2z, z = 2. (DS. ) 3 D25. z x2 + y 2dxdydz; y 2 = 3x − x2, z = 0, z = 2. (DS. 24) D
  • 145. 144 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e T´ thˆ t´ cua c´c vˆt thˆ gi´.i han bo.i c´c m˘t d˜ chı ra. ’ ınh e ıch ’ a a . ’ e o . ’ a a a ’ . 26. x = 0, y = 0, z = 0, x + 2y + z − 6 = 0. (DS. 36) 27. 2x + 3y + 4z = 12; x = 0, y = 0, z = 0. (DS. 12) x y z abc 28. + + = 1, x = 0, y = 0, z = 0. (DS. ) a b c 6 2 2 πa3 29. ax = y + z , x = a. (DS. ) 2 30. 2z = x2 + y 2, z = 2. (DS. 4π) π √ 31. z = x2 + y 2, x2 + y 2 + z 2 = 2. (DS. [8 2 − 7]) 6 π 32. z = x2 + y 2 , z = x2 + y 2 . (DS. ) 6 π 33. x2 + y 2 − z = 1, z = 0. (DS. ) 2 81π 34. 2z = x2 + y 2, y + z = 4. (DS. ) 4 x2 y 2 z 2 4 35. + + 2 = 1. (DS. πabc) a2 b2 c 3 12.3 T´ phˆn du.`.ng ıch a o 12.3.1 C´c dinh ngh˜ co. ban a . ıa ’ Gia su. h`m f(M), P (M) v` Q(M ), M = (x, y) liˆn tuc tai moi diˆm ’ ’ a a e . . . e ’ cua du.`.ng cong do du.o.c L = L(A, B) v´.i diˆm dˆu A v` diˆm cuˆi B. ’ o . o e `’ a a e ’ ´ o o a u ´ a .i dˆ d`i tu.o.ng u.ng ’ o . Chia mˆt c´ch t`y y L(A, B) th`nh n cung nho v´ o a ´ . a . ˜ l` ∆s0, ∆s1, ∆s2, . . . , ∆sn−1 . D˘t d = max (∆si ). Trong mˆ i cung a o 0 i n−1 ’ a ´ o a . u ´ e ’ nho, lˆy mˆt c´ch t`y y diˆm N0 , N1, . . . , Nn−1 . t´nh gi´ tri f (Ni ), ı a . a . e ’ P (Ni ) v` Q(Ni ) tai diˆm Ni d´. o X´t hai phu.o.ng ph´p lˆp tˆng t´ phˆn sau dˆy. e a a o . ’ ıch a a
  • 146. 12.3. T´ phˆn d u.`.ng ıch a o 145 Phu.o.ng ph´p I. Lˆy gi´ tri f(Ni ) nhˆn v´.i dˆ d`i cung ∆si tu.o.ng a ´ a a . a o o a .u´.ng v` lˆp tˆng t´ phˆn a a o ’ ıch a . n−1 σ1 = f (Ni )∆si . (*) i=0 Phu.o.ng ph´p II. Kh´c v´.i c´ch lˆp tˆng t´ch phˆn (∗), trong a a o a a o . ’ ı aphu.o.ng ph´p n`y ta lˆy gi´ tri P (Ni ), Q(Ni ) nhˆn khˆng phai v´.i a a ´ a a . a o ’ o o a ’ a ’ a a a odˆ d`i cua c´c cung nho m` l` nhˆn v´ ı.i h`nh chiˆu vuˆng g´c cua c´c ´ e o o ’ a .cung nho d´ trˆn c´c truc toa dˆ, t´.c l` lˆp tˆng ’ o e a . . o u a a o . . ’ n−1 σx = P (Ni )∆xi; ∆xi = proOx ∆si, i=0 n−1 σy = Q(Ni )∆yi; ∆yi = proOy ∆si . i=0 ˜ a o . ’ ıch a e a e ˜ a ´ e o . ’ Mˆ i c´ch lˆp tˆng t´ phˆn trˆn dˆy s˜ dˆ n dˆn mˆt kiˆu t´ch o a e ıphˆn du.`.ng. a oDinh ngh˜ 12.3.1. Nˆu tˆn tai gi´.i han h˜.u han lim σ1 khˆng phu-. ıa ´ o e ` . o . u . d→0 o .thuˆc v`o ph´p phˆn hoach du o o a e a .`.ng cong L th`nh c´c cung nho v` a a ’ a . . o . o a . e . . a e’ e ˜khˆng phu thuˆc v`o viˆc chon c´c diˆm trung gian Ni trˆn mˆ i cung onho th` gi´.i han d´ du.o.c goi l` t´ phˆn du.`.ng theo dˆ d`i (hay t´ ’ ı o . o . . a ıch a o o a . ıch a .`.ng kiˆu I) cua h`m f (x, y) theo du.`.ng cong L = L(A, B).phˆn du o e’ ’ a oK´ hiˆu: y e . f(x, y)ds. (12.23) LDinh ngh˜ 12.3.2. Ph´t biˆu tu.o.ng tu. nhu. trong dinh ngh˜ 12.3.1:-. ıa a e ’ . . ıa n−1 +1 . lim σx = lim P (Ni )∆xi = P (x, y)dx d→0 d→0 i=0 L(A,B) (12.24)
  • 147. 146 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e goi l` t´ phˆn du.`.ng theo ho`nh dˆ (nˆu (12.24) tˆn tai h˜.u han) . a ıch a o a . ´ o e ` . u o . n−1 2+ . lim σy = lim Q(Ni )∆yi = Q(x, y)dy d→0 d→0 i=0 L(A,B) (12.25) goi l` t´ phˆn du.`.ng theo tung dˆ (nˆu (12.25) tˆn tai h˜.u han) . a ıch a o . ´ o e ` . u o . o .`.ng ngu.`.i ta lˆp tˆng t´ phˆn dang Thˆng thu o o a o ’ ıch a . . n−1 n−1 Σ= P (Ni )∆xi + Q(Ni )∆yi i=0 o=0 v` nˆu ∃ lim Σ th` gi´.i han d´ du.o.c goi l` t´ phˆn du.`.ng theo toa a e´ ı o . o . . a ıch a o . d→0 . o’ dˆ dang tˆng qu´t: o . a P (x, y)dx + Q(x, y)dy. (12.26) L(A,B) Dinh l´. Nˆu c´c h`m f(x, y), P (x, y), Q(x, y) liˆn tuc theo du.`.ng -. y ´ e a a e . o cong L(A, B) = L th` c´c t´ phˆn du.`.ng (12.23) - (12.26) tˆn tai ı a ıch a o ` . o h˜ u.u han. . T`. dinh ngh˜ 12.3.1 v` kh´i niˆm dˆ d`i cung (khˆng phu thuˆc u . ıa a a e . o a . o . o . .´.ng cua cung) v` dinh ngh˜a 12.3.2 v` t´ chˆt cua h` chiˆu cua hu o ’ a . ı ´ a ınh a ’ ınh ´ e ’ ´ ’ ´ cung (h` chiˆu dˆi dˆu khi dˆi hu o ınh e o a o’ .´.ng cua cung) suy ra t´nh chˆt ’ ı ´ a quan trong cua t´ phˆn du.`.ng: t´ch phˆn du.`.ng theo dˆ d`i khˆng . ’ ıch a o ı a o o a . o phu thuˆc v`o hu o o a .´.ng cua du.`.ng cong; t´ phˆn du.`.ng theo toa dˆ ’ o ıch a o o . . . . ’ ´ ’ dˆi dˆu khi dˆi hu o o a o .´.ng du.`.ng cong. o 12.3.2 T´ t´ phˆn du.`.ng ınh ıch a o Phu.o.ng ph´p chung dˆ t´nh t´ phˆn du.`.ng l` du.a viˆc t´ t´ a ’ e ı ıch a o a e ınh ıch . .`.ng vˆ t´ phˆn x´c dinh. Cu thˆ l`: xuˆt ph´t t`. phu.o.ng phˆn du o a ` ıch a a . e ’ a ´ . e a a u
  • 148. 12.3. T´ phˆn d u.`.ng ıch a o 147tr` cua du.`.ng lˆy t´ phˆn L = L(A, B) ta biˆn dˆi biˆu th´.c du.´.i ınh ’ o ´ a ıch a ´ ’ e o e ’ u o ´dˆu t´ phˆn du o a ıch a .`.ng th`nh biˆu th´.c mˆt biˆn m` gi´ tri cua biˆn d´ a ’ e u o e ´ a a . ’ ´ e o . . e ’ ` a a e ’ ´ o e a a ’ ıch a a .tai diˆm dˆu A v` diˆm cuˆi B s˜ l` cˆn cua t´ phˆn x´c dinh thu . .o.c.du . 1+ Nˆu L(A, B) du.o.c cho bo.i c´c phu.o.ng tr` tham sˆ x = ϕ(t), ´ e . ’ a ınh ´ oy = ψ(t), t ∈ [a, b] (trong d´ ϕ, ψ kha vi liˆn tuc v` ϕ + ψ 2 > 0) th` o ’ e . a 2 ı ds = ϕ 2 + ψ 2 dt b f(x, y)ds = f[ϕ(t), ψ(t)] ϕ 2 + ψ 2 dt (12.27) L(A,B) av` a P (x, y)dx + Q(x, y)dy = L(A,B) b = P ϕ(t), ψ(t) ϕ (t) + Q ϕ(t), ψ(t) ψ (t) dt. (12.28) a 2+ Nˆu L(A, B) du.o.c cho bo.i phu.o.ng tr`nh y = g(x), x ∈ [a, b] e´ . ’ ı ’(trong d´ g(x) kha vi liˆn tuc trˆn [a, b]) th` o e . e ı ds = 1 + g 2(x)dx b f(x, y)ds = f[x, g(x)] 1 + g 2(x)dx. (12.29) L(A,B) av` a b P dx + Qdy = P (x, g(x)) + Q(x, g(x))g (x) dx. (12.30) L(A,B) a
  • 149. 148 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 3+ Nˆu L(A, B) du.o.c cho du.´.i dang toa dˆ cu.c ρ = ρ(ϕ) α ´ e . o . . o . . ϕ β th` ı ds = ρ2 + ρϕ 2 dϕ β f(x, y)ds = f [ρ cos ϕ, ρ sin ϕ] ρ2 + ρ 2 dϕ. (12.31) L(A,B) α 4+ T´ phˆn du.`.ng theo toa dˆ c´ thˆ t´nh nh`. cˆng th´.c Green. ıch a o . o o e ı . ’ o o u ∂Q ∂P ´ Nˆu P (x, y), Q(x, y) v` c´c dao h`m riˆng e a a . a e , c`ng liˆn tuc u e . ∂x ∂y trong miˆn D gi´.i han bo.i du.`.ng cong khˆng tu. c˘t tro.n t`.ng kh´c `e o . ’ o o . a ´ u u L = ∂D th` ı ∂Q ∂P P dx + Qdy = − dxdy. (12.32) ∂x ∂y L+ D Cˆng th´.c (12.32) goi l` cˆng th´.c Green, trong d´ o u . a o u o l` t´ch phˆn a ı a L+ theo du.`.ng cong k´ c´ hu.´.ng du.o.ng L+ . o ın o o Hˆ qua. Diˆn t´ miˆn D gi´.i han bo.i du.`.ng cong L du.o.c t´ e . ’ e ıch ` . e o . ’ o . ınh theo cˆng th´ o u.c 1 SD = xdy − ydx. (12.33) 2 L 5+ Nhˆn x´t vˆ t´ch phˆn du.`.ng trong khˆng gian. Gia su. L = a e ` ı . e a o o ’ ’ L(A, B) l` du.`.ng cong khˆng gian; f, P, Q, R l` nh˜.ng h`m ba biˆn a o o a u a ´ e liˆn tuc trˆn L. Khi d´ tu.o.ng tu. nhu. tru.`.ng ho.p du.`.ng cong ph˘ng e . e o . o . o ’ a ta c´ thˆ dinh ngh˜ t´ phˆn du.`.ng theo dˆ d`i o e .’ ıa ıch a o o a . f (x, y, z)ds v` a L(A,B) t´ phˆn du.`.ng theo toa dˆ ıch a o . o . P (x, y, z)dx, Q(x, y, z)dy, R(x, y, z)dz L L L
  • 150. 12.3. T´ phˆn d u.`.ng ıch a o 149v` a P dx + Qdy + Rdz. L Vˆ thu.c chˆt k˜ thuˆt t´ c´c t´ch phˆn n`y khˆng kh´c biˆt g` ` . e ´ a y a ınh a ı . a a o a e ı .so v´ o.i tru.`.ng ho.p du.`.ng cong ph˘ng. o o ’ a . CAC V´ DU ´ I . xV´ du 1. T´ t´ phˆn du.`.ng ı . ınh ıch a o ds, trong d´ L l` cung parabˆn o a o y √ Ly 2 = 2x t`. diˆm (1, 2) dˆn diˆm (2, 2). u e ’ ´ e ’ e ’ Giai. Ta t` vi phˆn dˆ d`i cung. Ta c´ ım a o a. o √ 1 y = 2x, y = √ , 2x √ 2 1 1 + 2x ds = 1 + y dx = 1 + dx = √ dx. 2x 2xTu. d´ suy ra o 2 √ x x 1 + 2x 1 √ √ ds = √ · √ dx = [5 5 − 3 3]. y 2x 2x 6 L 1V´ du 2. T´ dˆ d`i cua du.`.ng astroid x = a cos3 t, y = a sin3 t, ı . ınh o a ’ . ot ∈ [0, 2π]. Giai. Ta ´p dung cˆng th´.c: dˆ d`i (L) = ds. Trong tru.`.ng ’ a . o u o a . o Lho.p n`y ta c´ . a o 3a x = −3a cos2 t sin t, y = 3a sin2 t cos t, ds = sin 2tdt. 2V` du.`.ng cong dˆi x´.ng v´.i c´c truc toa dˆ nˆn ı o ´ o u o a . . o e . π/2 3a − cos 2t π/2 dˆ d`i(L) = 4 o a . sin 2tdt = 6a = 6a. 2 2 0 0
  • 151. 150 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e V´ du 3. T´ ı . ınh (x − y)ds, trong d´ L : x2 + y 2 = 2ax. o L Giai. Chuyˆn sang toa dˆ cu.c x = r cos ϕ, y = r sin ϕ. Trong toa ’ e’ . o .. . dˆ cu o . .c phu.o.ng tr` du.`.ng tr`n c´ dang r = 2a cos ϕ, − π ϕ π . ınh o o o . . 2 2 Vi phˆn dˆ d`i cung a o a . ds = r2 + rϕ 2 dϕ = 4a2 cos2 ϕ + 4a2 sin2 ϕdϕ = 2adϕ. o Do d´ π/2 I= (x − y)ds = (2a cos ϕ) cos ϕ − (2a sin ϕ) sin ϕ 2adϕ L −π/2 π/2 = 4a2 cos2 ϕdϕ = 2πa2 . −π/2 V´ du 4. T´ t´ phˆn ı . ınh ıch a (3x2 + y)dx + (x − 2y 2 )dy, trong d´ L l` o a L biˆn cua h` tam gi´c v´.i dınh A(0, 0), B(1, 0), C(0, 1). e ’ ınh a o ’ Giai. Theo t´ chˆt cua t´ch phˆn du.`.ng ta c´ ’ ´ ınh a ’ ı a o o = + + . L AB BC CA a) Trˆn canh AB ta c´ y = 0 ⇒ dy = 0. Do d´ e . o o 1 = 3x2dx = 1. AB 0 b) Trˆn canh BC ta c´ x + y = 1 ⇒ y = −x + 1, dy = −dx. Do d´ e . o o 0 5 = [3x2 + (1 − x) − x + 2(1 − x2)]dx = − · 3 BC 1
  • 152. 12.3. T´ phˆn d u.`.ng ıch a o 151 c) Trˆn canh CA ta c´ x = 0 ⇒ dx = 0 v` do d´ e . o a o 0 2 =− 2y 2dy = · 3 CA 1Nhu. vˆy a . 5 2 =1− + = 0. 3 3 LV´ du 5. T´ t´ phˆn ı . ınh ıch a (x +y)dx −(x − y)dy, trong d´ L l` du.`.ng o a o L x2 y 2elip 2 + 2 = 1 c´ dinh hu.´.ng du.o.ng. o . o a b Giai. 1 Ta c´ thˆ t´ tru.c tiˆp t´ch phˆn d˜ cho b˘ng c´c phu.o.ng ’ + ’ o e ınh . ´ e ı a a ` a aph´p d˜ nˆu (ch˘ng han b˘ng c´ch tham sˆ h´a phu.o.ng tr` elip). a a e ’ a . ` a a ´ o o ınh + 2 Nhu .ng do.n gian ho.n ca l` su. dung cˆng th´.c Green. Ta c´ ’ ’ a ’ . o u o ∂Q ∂P P = x + y, Q = −(x − y) ⇒ − = −2. ∂x ∂yDo d´ theo cˆng th´.c Green ta c´ o o u o = (−2)dxdy = −2πab, L x2 2 + y2 1 a2 b ı e ıch ınh . `v` diˆn t´ h` elip b˘ng πab. aV´ du 6. T´ t´ phˆn ı . ınh ıch a 2(x2 + y 2)dx + x(4y + 3)dy, trong d´ L l` o a Ldu.`.ng gˆp kh´c ABC v´.i dınh A(0, 0), B(1, 1) v` C(0, 2). o ´ a u o ’ a ’ ´ e ´ Giai. Nˆu ta nˆi A v´ o o.i C th` thu du.o.c du.`.ng gˆp kh´c k´n L∗ ı o ´ a u ı . .i han ∆ABC. Trˆn canh CA ta c´ x = 0 nˆn dx = 0 v` t`. d´gi´ . o e . o e a u o 2(x2 + y 2 )dx + x(4y + 3)dy = 0. CA
  • 153. 152 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e Do d´ o + = ⇒ = . L CA L∗ L L∗ Ap dung cˆng th´.c Green ta c´ ´ . o u o = [(4y + 3) − 4y]dxdy = 3 dxdy L ∆ABC ∆ABC = 3S∆ABC = 3. ` ˆ BAI TAP . T´ c´c t´ phˆn du.`.ng theo dˆ d`i sau dˆy ınh a ıch a o o a . a √ 1. (x + y)ds, C l` doa n th˘ng nˆi A(9, 6) v´.i B(1, 2). (DS. 36 5) a . ’ a ´ o o C 2. xyds, C l` biˆn h` vuˆng |x| + |y| = a, a > 0. (DS. 0) a e ınh o C 3. a e ’ a ’ (x + y)ds, C l` biˆn cua tam gi´c dınh A(1, 0), B(0, 1), C(0, 0). C √ (DS. 1 + 2) ds √ 4. , C l` doan th˘ng nˆi A(0, 2) v´.i B(4, 0). (DS. 5 ln 2) a . ’ a ´ o o x−y C 5. x2 + y 2ds, C l` du.`.ng tr`n x2 + y 2 = ax. a o o (DS. 2a2 ) C 6. (x2 + y 2)n ds, C l` du.`.ng tr`n x2 + y 2 = a2 . a o o (DS. 2πa2n+1) C √ x2 +y 2 7. e ds, C l` biˆn h`nh quat tr`n a e ı . o C
  • 154. 12.3. T´ phˆn d u.`.ng ıch a o 153 π (r, ϕ) : 0 r a, 0 ϕ . 4 πaea (DS. 2(ea − 1) + ) 48. xyds, C l` mˆt phˆn tu. elip n˘m trong g´c phˆn tu. I. a o. ` a ` a o ` a C ab a2 + ab + b2 (DS. · ) 3 a+b Chı dˆ n. Su. dung phu.o.ng tr`nh tham sˆ cua du.`.ng elip: x = ’ a˜ ’ . ı ´ o ’ oa cos t, y = b sin t. ds9. , C l` doan th˘ng nˆi diˆm O(0, 0) vo.i A(1, 2). a . ’ a ´ ’ o e x 2 + y2 + 4 C √ 5+3 (DS. ln ) 410. (x2 + y 2 + z 2 )ds, C l` cung du.`.ng cong x = a cos t, y = a sin t, a o Cz = bt; 0 t 2π, a > 0, b > 0. 2π √ 2 (DS. a + b2 (3a2 + 4π 2b2 )) 311. x2ds, C l` du.`.ng tr`n a o o C  x2 + y 2 + z 2 = a2 2πa3 (DS. ) x + y + z = 0 3 Chı dˆ n. Ch´.ng to r˘ng ’ a˜ u ’ a` x2 ds = y 2ds = z 2 ds v` t`. d´ suy a u o C C Cra 1 I= (x2 + y 2 + z 2)ds. 3 C
  • 155. 154 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 12. (x + y)ds, C l` mˆt phˆn tu. du.`.ng tr`n a o. ` a o o C  x2 + y 2 + z 2 = R2 y = x √ n˘m trong g´c phˆn t´m I. (DS. R2 2) ` a o ` a a 13. T´ ınh xyzds, C l` mˆt phˆn tu. du.`.ng tr`n a o. ` a o o C  x2 + y 2 + z 2 = R2 2  x2 + y 2 = R 4 ` a o ` a n˘m trong g´c phˆn t´m I. a T´ c´c t´ phˆn du.`.ng theo toa dˆ sau dˆy ınh a ıch a o . o . a 14. y 2 dx + x2dy, C l` du.`.ng t`. diˆm (0, 0) dˆn diˆm (1, 1): a o u e ’ ´ e ’ e C a . ’ 1) C l` doan th˘ng. a 2) C l` cung parabol y = x2. a √ 3) C l` cung parabol y = x. a 2 7 7 (DS. 1) ; 2) ; 3) ) 3 10 10 15. y 2 dx − x2 dy, C l` du.`.ng tr`n b´n k´nh R = 1 v` c´ hu.´.ng a o o a ı a o o C ngu.o.c chiˆu kim dˆng hˆ v`: . `e ` o ` a o 1) v´ a o.i tˆm tai gˆc toa dˆ. ´ . o . o . 2) v´ a o.i tˆm tai diˆm (1, 1). ’ . e (DS. 1) 0; 2) −4π) 16. xdy − ydx, C l` du.`.ng gˆp kh´c dınh tai c´c diˆm (0, 0), (1, 0) a o ´ a u ’ . a ’ e C
  • 156. 12.3. T´ phˆn d u.`.ng ıch a o 155 v` (1, 2). a (DS. 2)17. cos ydx − sin xdy, C l` doa n th˘ng t`. diˆm (2, −2) dˆn diˆm a . ’ a u e ’ ´ e ’ e C (−2, 2). (DS. −2 sin 2)18. (x2 + y 2)dx + (x2 − y 2)dy, C l` du.`.ng cong y = 1 − |1 − x|, a o C 4 0 x 2. (DS. ) 3 x2 y 219. (x + y)dx + (x − y)dy, C l` elip c´ hu.´.ng du.o.ng 2 + 2 = 1. a o o a b C (DS. 0)20. (2a − y)dx + xdy, C l` mˆt v`m cuˆn cua du.`.ng xicloid a o o . ´ o ’ o C x = a(t − sin t), y = a(1 − cos t), 0 t 2π. (DS. −2πa2) dx + dy21. , C l` biˆn c´ hu.o.ng du.o.ng cua h`nh vuˆng v´.i dınh a e o ´ ’ ı o o ’ |z| + |y| C . e ’ tai diˆm A(1, 0), B(0, 1), C(−1, 0) v` D(0, −1). a (DS. 0)22. (x2 − y 2)dx + (x2 + y 2)dy, C l` elip c´ hu.´.ng du.o.ng a o o C x2 y 2 + = 1. (DS. 0) a2 b223. (x2 + y 2)dx + xydy, C l` cung cua du.`.ng y = ex t`. diˆm a ’ o u e ’ C 3e2 1 ´ e e’ (0, 1) dˆn diˆm (1, e). (DS. + ) 4 224. (x3 − y 2)dx + xydy, C l` cung cua du.`.ng y = ax t`. diˆm a ’ o u e ’ C 1 a2 3(1 − a2) ´ e’ (0, 1) dˆn diˆm (1, a). e (DS. + + ) 4 2 4 ln a25. y 2dx + x2 dy, C l` v`m th´. nhˆt cua du.`.ng xicloid a o u ´ a ’ o C
  • 157. 156 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e x = a(t − sin t), y = a(1 − cos t), a > 0 c´ dinh hu.´.ng theo hu.´.ng o . o o t˘ng cua tham sˆ. (DS. a3 π(5 − 2π)) a ’ ´ o Ap dung cˆng th´.c Green dˆ t´nh t´ch phˆn du.`.ng ´ . o u ’ e ı ı a o πa4 26. xy 2dy − x2 dx, C l` du.`.ng tr`n x2 + y 2 = a2. (DS. a o o ) 4 C x2 y 2 27. (x + y)dx − (x − y)dy, C l` elip a + 2 = 1. (DS. −2πab) a2 b C (cos 2xydx + sin 2xydy), C l` du.`.ng tr`n x2 + y 2 = R2 . 2 +y 2 28. e−x a o o C (DS. 0) 29. (xy + ex sin x + x + y)dx + (xy − e−y + x − sin y)dy, C C l` du.`.ng tr`n x2 + y 2 = 2x. a o o (DS. −π) 30. (1 + xy)dx + y 2 dy, C l` biˆn cua nu.a trˆn cua h`nh tr`n a e ’ ’ e ’ ı o C π x + y2 2 2x (y 0). (DS. − ) 2 31. (x2 + y 2)dx + (x2 − y 2)dy, C l` biˆn cua tam gi´c ∆ABC v´.i a e ’ a o C e’ ´ e ’ ` A = (0, 0), B = (1, 0), C = (0, 1), Kiˆm tra kˆt qua b˘ng c´ch a a t´ tru ınh . .c tiˆp. (DS. 0) ´ e 32. (2xy − x2)dx + (x + y 3)dy, C l` biˆn cua miˆn bi ch˘n gi´.i han a e ’ ` . a e . o . C bo.i hai du.`.ng y = x2 v` y 2 = x. Kiˆm tra kˆt qua b˘ng c´ch t´nh ’ o a ’ e ´ e ’ `a a ı .c tiˆp. (DS. 1 tru. ´ e ) 30 33. ex [(1 − cos y)dx − (y − sin y)dy], C l` biˆn cua tam gi´c ABC a e ’ a C v´.i A = (1, 1), B = (0, 2) v` C = (0, 0). (DS. 2(2 − e)) o a
  • 158. 12.3. T´ phˆn d u.`.ng ıch a o 15734. (xy + x + y)dx + (xy + x − y)dy, trong d´ C l` o a C x2 y 2 a) elip 2 + 2 = 1; a b 3 .`.ng tr`n x2 + y 2 = ax (a > 0). (DS. a) 0; b) − πa ) b) du o o 8 πR435. xy 2dx − x2ydy, C l` du.`.ng tr`n x2 + y 2 = R2 . (DS. a o o ) 2 C36. 2(x2 + y 2 )dx + x(4y + 3)dy, C l` du.`.ng gˆp kh´c v´.i dınh a o ´ a u o ’ C ’ e ´ e ’ `A = (0, 0), B = (1, 1), C = (0, 2). Kiˆm tra kˆt qua b˘ng c´ch t´nh a a ıtru.c tiˆp. (DS. 3) . e´ Chı dˆ n. Bˆ sung cho C doan th˘ng dˆ thu du.o.c chu tuyˆn d´ng. ’ a˜ o’ . ’ a ’ e . ´ e o37. H˜y so s´nh hai t´ phˆn a a ıch aI1 = (x + y)2 dx − (x − y)2dy v` I2 = a (x + y)2dx − (x − y)2 dy AmB AnBnˆu AmB l` doa n th˘ng nˆi A(1, 1) v´.i B(2, 6) v` AnB l` cung parabol ´ e a . ’ a ´ o o a a ´qua A. B v` gˆc toa dˆ. (DS. I1 − I2 = 2) a o . o .38. T´ I = ınh (x + y)dx − (x − y)dy, trong d´ AmB l` cung o a AmBnAparabol qua A(1, 0) v` B(2, 3) v` c´ truc dˆi x´.ng l` truc Oy, c`n a a o . o u ´ a . oAnB l` doa n th˘ng nˆi A v´.i B. a . ’ a ´ o o 1 (DS. − ) 3 Chı dˆ n. Dˆu tiˆn viˆt phu.o.ng tr` parabol v` du.`.ng th˘ng, sau ’ a˜ ` a e ´ e ınh a o ’ ad´ ´p dung cˆng th´.c Green. oa . o u39. Ch´.ng minh r˘ng gi´ tri cua t´ phˆn u ` a a . ’ ıch a (2xy − y)dx + x2dy, Ctrong d´ C l` chu tuyˆn d´ng, b˘ng diˆn t´ miˆn ph˘ng v´.i biˆn l` o a ´ e o ` a e ıch ` . e ’ a o e aC.
  • 159. 158 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 40. (x + y)2dx − (x2 + y 2)dy, C l` biˆn cua ∆ABC v´.i dınh a e ’ o ’ C 2 A(1, 1), B(3, 2) v` C(2, 5). (DS. −46 ) a 3 41. (y − x2)dx + (x + y 2)dy, C l` biˆn h`nh quat b´n k´ R v` a e ı . a ınh a C π g´c ϕ (0 o ϕ ). (DS. 0) 2 42. y 2 dx + (x + y)2dy, C l` biˆn cua h`nh tam gi´c ∆ABC v´.i a e ’ ı a o C 2a3 A(a, 0), B(a, a), C(0, a). (DS. ) 3 12.4 T´ phˆn m˘t ıch a a . 12.4.1 C´c dinh ngh˜ co. ban a . ıa ’ Gia su. c´c h`m f (M), P (M), Q(M) v` R(M ), M = (x, y, z) liˆn tuc ’ ’ a a a e . ’ ’ tai moi diˆm M cua m˘t tro .n, do du.o.c (σ) (m˘t tro.n l` m˘t c´ m˘t . . e a . . a . a a o a . . ph˘a’ ng tiˆp x´c tai moi diˆm cua n´). Chia mˆt c´ch t`y y m˘t (σ) e´ u . . ’ e ’ o o a . u ´ a . ’ th`nh n manh con σ0 , σ1, . . . , σn−1 v´ a o .i diˆn t´ tu.o.ng u.ng l` ∆S0, e ıch ´ a . ∆S1 , . . . , ∆Sn−1 . D˘t dk = diamσk ; d = max dk . Trong mˆ i manh a . ˜ o ’ 0 k n−1 . ´ o a . u ´ e ’ ı a . ’ a a m˘t ta lˆy mˆt c´ch t`y y diˆm Ni . T´nh gi´ tri cua c´c h`m d˜ cho a a a ’m Ni , i = 0, n − 1. Ta k´ hiˆu cos α(Ni ), cos β(Ni ) v` cos γ(Ni ) tai diˆ . e y e . a ’ l` c´c cosin chı phu a a .o.ng cua vecto. ph´p tuyˆn n(Ni ) tai diˆm Ni cua ’ a ´ e ’ ’ . e m˘t (σ). a . a a o . ’ X´t hai c´ch lˆp tˆng t´ phˆn sau. e ıch a (I) Lˆy gi´ tri f(Ni ) nhˆn v´.i c´c phˆn tu. diˆn t´ch m˘t ∆S0, ´ a a . a o a ` a ’ e ı. a . ∆S1 , . . . , ∆Sn−1 v` lˆp tˆ a a o . ’ng n−1 f (Ni )δSi i=0
  • 160. 12.4. T´ phˆn m˘t ıch a a . 159 (II) Kh´c v´.i c´ch lˆp tˆng t´ phˆn trong (I), trong phu.o.ng ph´p a o a a o . ’ ıch a a ´ a a . a a o ’ on`y ta lˆy gi´ tri P (Ni ), Q(Ni ) v` R(Ni ) nhˆn khˆng phai v´ a .i phˆn ` atu. diˆn t´ ∆Si cua c´c manh m˘t σi m` l` nhˆn v´.i h`nh chiˆu cua ’ e ıch . ’ a ’ a . a a a o ı ´ e ’c´c manh d´ lˆn c´c m˘t ph˘ng toa dˆ Oxy, Oxz v` Oyz, t´.c l` lˆp a ’ o e a a. ’ a . o. a u a a . a o ’c´c tˆng dang . n−1 i i σxy = P (Ni )m(σxy ), m(σxy ) = proOxy (σi); i=0 n−1 i i σxz = Q(Ni)m(σxz ), m(σxz ) = proOxz (σi); i=0 n−1 i i σyz = R(Ni )m(σyz ), m(σyz ) = proOyz (σi ). i=0D.nh ngh˜ 12.4.1. Nˆu tˆn tai gi´.i han h˜.u han-i ıa ´ o e ` . o . u . n−1 lim f (Ni )∆Si (12.34) d→0 i=1 . o a . e a . a . a a ’khˆng phu thuˆc v`o ph´p phˆn hoach m˘t (σ) th`nh c´c manh con o a o . o a a . . a e ’v` khˆng phu thuˆc v`o c´ch chon c´c diˆm trung gian Ni ∈ σi th` ıgi´.i han d´ goi l` t´ phˆn m˘t theo diˆn t´ o . o . a ıch a a . e ıch. . K´ hiˆu : y e . f (x, y, z)dS. (σ)Dinh ngh˜ 12.4.2. C´c t´ phˆn m˘t theo toa dˆ du.o.c dinh ngh˜a-. ıa a ıch a a . . o . . . ı ’bo.i n−1 def i P (M)dxdy = lim P (Ni )m(σxy ) (12.35) d→0 i=0 (σ) n−1 def i Q(M)dxdz = lim Q(Ni )m(σxz ) (12.36) d→0 i=0 (σ) n−1 def i R(M )dydz = lim R(Ni )m(σyz ) (12.37) d→0 i=0 (σ)
  • 161. 160 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e nˆu c´c gi´.i han o. vˆ phai (12.35)-(12.37) tˆn tai h˜.u han khˆng phu ´ e a o . ’ e ’ ´ ` . u o . o . . e a . a. a a . ’ thuˆc v`o ph´p phˆn hoach m˘t (σ) v` c´ch chon diˆm trung gian Ni , o a e i = 0, n − 1. ıch a a . . o . . o’ T´ phˆn m˘t theo toa dˆ dang tˆng qu´t a P (M)dxdy + Q(M)dxdz + R(M)dydz (σ) ’ ’ a ıch a l` tˆng cua c´c t´ phˆn m˘t theo toa dˆ (12.35), (12.36) v` (12.37). a o a . . o . a ´ e a a o . ı ı ı a a . ıa a ’ Nˆu (σ) l` m˘t d´ng (k´n !) th` t´ch phˆn m˘t theo ph´ ngo`i cua n´ du.o.c k´ hiˆu o . y e . ho˘c do.n gian l` a . ’ a ´ nˆu n´i r˜ (σ) l` m˘t n`o; e o o a a a . (σ)+ (σ) c`n t´ phˆn theo ph´a trong du.o.c k´ hiˆu o ıch a ı . y e . ho˘c do.n gian l` a . ’ a (σ)− (σ) khi d˜ n´i r˜ (σ) l` m˘t n`o. a o o a a a . 12.4.2 Phu.o.ng ph´p t´ t´ phˆn m˘t a ınh ıch a a . Phu.o.ng ph´p chung dˆ t´nh t´ch phˆn m˘t ca hai dang l` du.a vˆ t´ch a ’ e ı ı a a ’ . . a ` ı e phˆn hai l´ a o.p. Cu thˆ l`: xuˆt ph´t t`. phu.o.ng tr`nh cua m˘t (σ) ta ’ ´ ’ . e a a a u ı a . ´ e o e ’ ’ biˆn dˆi biˆu th´ u.c du.´.i dˆu t´ch phˆn th`nh biˆu th´.c hai biˆn m` ´ o a ı a a e’ u e´ a miˆn biˆn thiˆn cua ch´ng l` h` chiˆu do.n tri cua (σ) lˆn m˘t ph˘ng `e e´ e ’ u a ınh ´ e . ’ e a. ’ a toa dˆ tu.o.ng u.ng v´.i c´c biˆn d´. . o . ´ o a ´ e o 1 Nˆu m˘t (σ) c´ phu.o.ng tr`nh z = ϕ(x, y) th` t´ch phˆn m˘t + e´ a . o ı ı ı a a . theo diˆn t´ du . e ıch .o.c biˆn dˆi th`nh t´ch phˆn hai l´.p theo cˆng th´.c ´ o a e ’ ı a o o u . dS = 1 + ϕx 2 + ϕy 2dxdy f(x, y, z)dS = f [x, y, ϕ(x, y)] 1 + ϕx 2 + ϕy 2dxdy (12.38) (σ) D(x,y) ´ e o o ’ e a . ’ trong d´ D(x, y) l` h` chiˆu vuˆng g´c cua (σ) lˆn m˘t ph˘ng Oxy. o a ınh a
  • 162. 12.4. T´ phˆn m˘t ıch a a . 161 Nˆu m˘t (σ) c´ phu.o.ng tr` y = ψ(x, z) th` ´ e a . o ınh ı f (x, y, z)dS = f[x, ψ(x, z), z] 1 + ψx2 + ψz 2dxdz, (12.39) (σ) D(x,z)trong d´ D(x, z) = proOxz (σ). o Nˆu m˘t (σ) c´ phu.o.ng tr` x = g(y, z) th` ´ e a . o ınh ı f(·)dS = f[g(y, z), y, z] 1 + gy 2 + gz 2dydz, (12.40) (σ) D(y,z)trong d´ D(y, z) = proOyz (σ). o 2 Gia thiˆt m˘t (σ) chiˆu du.o.c do.n tri lˆn c´c m˘t ph˘ng toa dˆ, + ’ ´ . e a ´ e . . e a a . ’ a . o . u.c l` m˘t c´ phu.o.ng tr` dangt´ a a o ınh . . z = ϕ(x, y), (x, y) ∈ D(x, y); y = ψ(x, z), (x, z) ∈ D(x, z); x = g(y, z), (y, z) ∈ D(y, z).Ta k´ hiˆu e1 , e2 , e3 l` c´c vecto. co. so. cua R3 v` cos α(M ) = cos(n, e1), y e . a a ’ ’ acos β(M ) = cos(n, e2 ), cos γ(M) = cos(n, e3). D´ l` c´c cosin chı o a a ’phu.o.ng cua vecto. ph´p tuyˆn v´.i m˘t (σ) tai diˆm M ∈ (σ). Khi d´ ’ a ´ e o a . . e ’ oc´c t´ phˆn m˘t theo toa dˆ lˆy theo m˘t hai ph´a du.o.c t´nh nhu. a ıch a a . . ´ . o a a . ı . ısau.  +  ´ P (x, y, ϕ(x, y))dxdy nˆu cos γ > 0; e     D(x,y) P (M)dxdy =  − (σ)   ´ P (x, y, ϕ(x, y))dxdy nˆu cos γ < 0 e   D(x,y)(t´.c l` dˆu “+” tu.o.ng u.ng v´.i ph´p t´ phˆn theo ph´ ngo`i (ph´ u a a ´ ´ o e ıch a ıa a ıa ’ a o a ´trˆn) cua m˘t, c`n dˆu “−” tu e .o.ng u.ng v´.i ph´p t´ phˆn theo ph´ ´ o e ıch a ıa .trong (ph´ du.´.i) cua m˘t. ıa o ’ a.
  • 163. 162 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e Tu.o.ng tu. ta c´ . o  +  ´ Q(x, ψ(x, z), z)dxdz nˆu cos β > 0, e     D(x,z) Q(M)dxdz =  − (σ)   Q(·)dxdz ´ nˆu cos β < 0; e   D(x,z)  +  R(g(y, z), y, z)dydz ´ nˆu cos α > 0 e     D(y,z) R(M )dydz =  − (σ)   R(·)dydz ´ nˆu cos α < 0. e   D(y,z) Nhˆn x´t. T´ phˆn m˘t theo toa dˆ lˆy theo phˆn m˘t tru v´.i du.`.ng a e . ıch a a . . ´ . o a ` a a . o . o sinh song song v´ o.i truc Oz l` b˘ng 0. Trong c´c tru.`.ng ho.p tu.o.ng a a ` a o . . ., c´c t´ch phˆn m˘t theo toa dˆ x, z hay y, z c˜ng = 0. tu a ı a a o u . . . . 12.4.3 Cˆng th´.c Gauss-Ostrogradski o u D´ l` cˆng th´.c o a o u ∂P ∂Q ∂R + + dxdydz = P dydz + Qdxdz + Rdxdy. ∂x ∂y ∂z D ∂D N´ x´c lˆp mˆi liˆn hˆ gi˜.a t´ phˆn m˘t theo m˘t biˆn ∂D cua D o a a . ´ o e e u ıch a . a . a . e ’ v´.i t´ phˆn 3-l´.p lˆy theo miˆn D ⊂ R3 . o ıch a o a ´ ` e 12.4.4 Cˆng th´.c Stokes o u D´ l` cˆng th´.c o a o u ∂Q ∂P ∂R ∂Q P dx + Qdy + Rdz = − dxdy + − dydz ∂x ∂y ∂y ∂z L (σ) ∂P ∂R + − dzdx. ∂z ∂x
  • 164. 12.4. T´ phˆn m˘t ıch a a . 163N´ x´c lˆp mˆi liˆn hˆ gi˜.a t´ phˆn m˘t theo m˘t (σ) v´.i t´ phˆn o a a . ´ o e e u ıch a . a. a . o ıch a .`.ng lˆy theo b`. L cua m˘t (σ).du o a´ o ’ a . Ta nhˆn x´t r˘ng sˆ hang th´. nhˆt o. vˆ phai cua cˆng th´.c Stokes a e a . ` ´ o . ´ ´ u a ’ e ’ ’ o u ´ ınh a e ’ oc˜ng ch´ l` vˆ phai cˆng th´ u u.c Green. Hai sˆ hang c`n lai thu du.o.c ´ o . o . . u. d´ bo.i ph´p ho´n vi tuˆn ho`n c´c biˆn x, y, z v` c´c h`m P, Q, R:t` o ’ e a . a ` a a ´ e a a a x P z ←− y R ←− Q CAC V´ DU ´ I .V´ du 1. T´ t´ phˆn ı . ınh ıch a a ` (6x + 4y + 3z)dS, trong d´ (σ) l` phˆn o a (σ)m˘t ph˘ng x + 2y + 3z = 6 n˘m trong g´c phˆn t´m th´. nhˆt. a. ’ a ` a o ` a a u ´ a ’ Giai. M˘t t´ phˆn l` tam gi´c ABC tronng d´ A(6, 0, 0), a ıch . a a a oB(0, 3, 0) v` C(0, 0, 2). Su. dung phu.o.ng tr`nh cua (σ) dˆ biˆn dˆi a ’ . ı ’ ’ ´ e e o ’t´ phˆn m˘t th`nh t´ phˆn 2-l´.p. T`. phu.o.ng tr`nh cua (σ) r´t ıch a a. a ıch a o u ı ’ u 1ra z = (6 − x − 2y). T`. d´ u o 3 √ 2 2 14 dS = 1 + zx + zy dxdy = dxdy. 2Do d´ o √ 14 3 I= [(6x + 4y + (6 − x − 2y)]dxdy 3 3 ∆OAB √ 3 6−2y 14 = dy (5x + 2y + 6)dx 3 0 0 √ 3 14 5 2 6−2y √ = x + 2xy + 6x dy = 54 14. 3 2 0 0
  • 165. 164 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e V´ du 2. T´ ı . ınh a ` 1 + 4x2 + 4y 2 dS, (σ) l` phˆn paraboloid tr`n a o (σ) ` a e a. ’ xoay z = 1 − x2 − y 2 n˘m trˆn m˘t ph˘ng Oxy. a Giai. M˘t (σ) chiˆu du.o.c do.n tri lˆn m˘t ph˘ng Oxy v` h` tr`n ’ a . ´ e . . e a . ’ a a ınh o 2 2 x +y 1 l` h`nh chiˆu cua n´: D(x, y) = (x, y) : x + y 2 1 . Ta a ı ´ e ’ o 2 t´ dS. Ta c´ zx = −2x, zy = −2y ⇒ dS = ınh o 1 + 4x2 + 4y 2dxdy. Do vˆy a . = 1 + 4x2 + 4y 2 · 1 + 4x2 + 4y 2dxdy (σ) D(x,y) = (1 + 4x2 + 4y 2 )dxdy. x2 +y 2 1 B˘ng c´ch chuyˆn sang toa dˆ cu.c ta c´ ` a a ’ e . o . . o 2π 1 I= dϕ (1 + 4r2 )rdr = 3π. 0 0 V´ du 3. T´ t´ phˆn ı . ınh ıch a (y 2 + z 2dxdy, trong d´ (σ) l` ph´a ngo`i o a ı a (σ) √ ’ a m˘t z = 1 − x2 gi´.i han bo.i c´c m˘t ph˘ng y = 0, y = 1. cu a. o . ’ a a . ’ a Giai. M˘t (σ) l` nu.a trˆn cua m˘t tru x2 + z 2 = 1, z 0. Do d´ ’ a. a ’ e ’ a . . o ınh ´ e ’ e a ’ h` chiˆu cua (σ) lˆn m˘t ph˘ng Oxy l` h`nh ch˜ a a ı . nhˆt x´c dinh bo.i u a a . ’ . . √ a ` c´c diˆu kiˆn: −1 e e . x 1, 0 y 1. Do d´ v` z = 1 − x2 nˆn o ı e cos γ > 0 v` a √ (y 2 + z 2)dxdy = [y 2 + ( 1 − x2)2 ]dxdy (σ) D(x,y) 1 1 = dx (y 2 + 1 − x2)dy = 2. −1 0
  • 166. 12.4. T´ phˆn m˘t ıch a a . 165V´ du 4. T´ t´ phˆn ı . ınh ıch a 2dxdy + ydxdz − x2zdydz, trong d´ (σ) o (σ)l` ph´ trˆn cua phˆn elipxoid 4x2 + y 2 + 4z 2 = 1 n˘m trong g´c phˆn a ıa e ’ `a ` a o ` at´m I. a Giai. Ta viˆt t´ phˆn d˜ cho du.´.i dang ’ ´ e ıch a a o . I=2 dxdy + ydydz − x2zdydz. (σ) (σ) (σ)v` su. dung phu.o.ng tr` cua m˘t (σ) dˆ biˆn dˆi mˆ i t´ phˆn. Lu.u a ’ . ınh ’ a . ’ ´ ’ ˜ e e o o ıch a´ `y r˘ng cos α > 0, cos β > 0, cos γ > 0. a (i) V` h` chiˆu cua m˘t (σ) lˆn m˘t ph˘ng Oxy l` phˆn tu. h`nh ı ınh ´ e ’ a . e a . ’ a a `a ı 2 2 x yelip 2 + 2 1 nˆn e 1 2 π I1 = dxdy = dxdy = (v` diˆn t´ elip = 2π) ı e ıch . 2 (σ) D(x,y) (ii) H` chiˆu cua (σ) lˆn m˘t ph˘ng Oxz l` phˆn tu. h`nh tr`n ınh ´ e ’ e a . ’ a a ` a ı o 24x + 4z 2 4⇔x +z 2 2 1. M˘t kh´c t` a a u . phu.o.ng tr`nh m˘t r´t ra ı a u . .y =2 1−x 2 − y 2 v` do d´ a o √I2 = ydxdz = 2 1 − x2 − z 2 dxdz = |chuyˆn sang toa dˆ cu.c| e’ . o . . (σ) D(x,y) π/2 1 √ π =2 dϕ 1 − r2 rdr = · 3 0 0 (iii) H` chiˆu cua (σ) lˆn m˘t ph˘ng Oyz l` mˆt phˆn tu. h`nh ınh ´ e ’ e a. ’ a a o . ` a ı 2 yelip + z2 1 (y 0, z 0). T`. phu.o.ng tr`nh m˘t (σ) r´t ra u ı a . u 4
  • 167. 166 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e y2 x= 1− − z 2 rˆi thˆ v`o h`m du.´.i dˆu t´ phˆn cua I3: ` o ´ e a a ´ o a ıch a ’ 4 y2 I3 = x2 zdydz = z 1− − z 2 dydz 4 (σ) D(y,z) √ 1 2 1−z2 y2 4 = dz z 1− − z 2 dy = · · · = · 4 15 0 0 4π 4 Nhu. vˆy I = 2I1 + I2 − I3 = a . − · 3 15 V´ du 5. T´ ı . ınh ydydz, trong d´ (σ) l` m˘t cua t´. diˆn gi´.i han o a a ’ u e . . o . (σ)− bo.i m˘t ph˘ng x + y + z = 1 v` c´c m˘t ph˘ng toa dˆ, t´ phˆn du.o.c ’ a . ’ a a a a. a’ . o ıch a . . ´y theo ph´ trong cua t´. diˆn. lˆ a ıa ’ u e . ’ . ’ a ´ Giai. M˘t ph˘ng x + y + z = 1 c˘t c´c truc toa dˆ tai A(1, 0, 0), a a a . . o .. B(0, 1, 0) v` C = (0, 0, 1). Ta k´ hiˆu gˆc toa dˆ l` O(0, 0, 0). T`. d´ a ´ y e o . o a . . u o suy ra m˘t k´ (σ) gˆm t`. 4 h`nh tam gi´c ∆ABC, ∆BCO, ∆ACO a ın . ` o u ı a a a ıch a a . a o’ ´ ’ o ıch a v` ∆ABO. Do vˆy t´ phˆn d˜ cho l` tˆng cua bˆn t´ phˆn. (i) T´ phˆn I1 = ıch a ydxdz. R´t y t`. phu.o.ng tr`nh m˘t (σ) ⊃ u u ı a. ABC ∆ABC ta c´ y = 1 − x − z v` do d´ o a o 1 1−x 1 I1 = − (1 − x − z)dxdz = dx (x + z − 1)dz = − · 6 ACO 0 0 (Lu.u y r˘ng cos β = cos(n, Oy) < 0 v` vecto. n lˆp v´.i hu.´.ng du.o.ng ´ a` ı a o . o truc Oy mˆt g´c t`, do d´ tru.´.c t´ch phˆn theo ∆ACO xuˆt hiˆn dˆu . o o u . o o ı a ´ . a e a ´ tr`.) u (ii) ydxdz = ydxdz = 0 (BCD) (ABO)
  • 168. 12.4. T´ phˆn m˘t ıch a a . 167v` m˘t ph˘ng BCO v` ABO dˆu vuˆng g´c v´.i m˘t ph˘ng Oxz. ı a. ’ a a ` e o o o a . ’ a (iii) ydxdz = 0dxdz = 0. (ACO) ACO 1Vˆy I = − . a . 6V´ du 6. T´ t´ phˆn I = ı . ınh ıch a x3 dydz + y 3dzdx + z 3 dxdy, trong (σ)d´ (σ) l` ph´ ngo`i m˘t cˆu x2 + y 2 + z 2 = R2 . o a ıa a a ` . a ’ ´ dung cˆng th´.c Gauss-Ostrogradski ta c´ Giai. Ap . o u o =3 (x2 + y 2 + z 2)dxdydz (σ) Dtrong d´ D ⊂ R3 l` miˆn v´.i biˆn l` m˘t (σ). Chuyˆn sang toa dˆ o a ` e o e a a . ’ e . o . `cˆu ta c´ a o 2π π R 3 (x2 + y 2 + z 2 )dxdydz = 3 dϕ sin θdθ r4 dr D 0 0 0 5 12πR = · 5 12πR5 Vˆy I = a . · 5V´ du 7. T´ t´ phˆn ı . ınh ıch a x2y 3dx + dy + zdz, trong d´ L l` du.`.ng o a o Ltr`n x2 + y 2 = 1, z = 0, c`n m˘t (σ) l` ph´ ngo`i cua nu.a m˘t cˆu o o a . a ıa a ’ ’ a ` . a 2 2 2x + y + z = 1, z > 0 v` L c´ dinh hu o a o . .´.ng du.o.ng. Giai. Trong tru.`.ng ho.p n`y P = x2 y 3, Q = 1, R = z. Do d´ ’ o . a o ∂Q ∂P ∂R ∂Q ∂P ∂R − = −3x2y 2, − = 0, − =0 ∂x ∂y ∂y ∂z ∂z ∂x
  • 169. 168 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e v` do d´ theo cˆng th´.c Stokes ta c´ a o o u o π = −3 x2y 2 dxdy = − · 8 L (σ) ` ˆ BAI TAP . T´ c´c t´ phˆn m˘t theo diˆn t´ sau dˆy ınh a ıch a a . e ıch . a 1. (x + y + z)dS, (Σ) l` m˘t lˆp phu.o.ng 0 a a a . . x 1, 0 1, (Σ) 0 z 1. (DS. 9) 2. a ` a a . ’ a ` (2x + y + z)dS, (Σ) l` phˆn m˘t ph˘ng x + y + z = 1 n˘m trong a (Σ) √ 2 3 o ` a g´c phˆn t´m I. (DS. a ) 3 4y 3. z + 2x + a ` ’ dS, (Σ) l` phˆn m˘t ph˘ng 6x + 4y + 3z = 12 a a . a 3 (Σ) √ `m trong g´c phˆn t´m I. (DS. 4 61) n˘ a o ` a a 4. x2 + y 2dS, (Σ) l` phˆn m˘t n´n z 2 = x2 + y 2, 0 a ` a a o . z 1. (σ) √ 2 2π (DS. ) 3 √ 5. (y + z + a2 − x2)dS, (Σ) l` phˆn m˘t tru x2 + y 2 = a2 n˘m a ` a a . . ` a (Σ) gi˜.a hai m˘t ph˘ng z = 0 v` z = h. (DS. ah(4a + πh)) u a . ’ a a 6. y 2 − x2dS, (Σ) l` phˆn m˘t n´n z 2 = x2 + y 2 n˘m trong m˘t a ` a a o . ` a a . (Σ) 8a3 tru x2 + y 2 = a2. . (DS. ) 3
  • 170. 12.4. T´ phˆn m˘t ıch a a . 1697. (x + y + z)dS, (Σ) l` nu.a trˆn cua m˘t cˆu x2 + y 2 + z 2 = a2. a ’ e ’ a ` . a (Σ) (DS. πa3) 8πa38. x2 + y 2dS, (Σ) l` m˘t cˆu x2 + y 2 + z 2 = a2 . (DS. a a ` . a ) 3 (Σ) dS9. , (Σ) l` biˆn cua t´. diˆn x´c dinh bo.i bˆt phu.o.ng a e ’ u e a . . ’ a ´ (1 + x + y) (Σ) 1 √ √tr` x+y+z ınh 1, x 0, y 0, z 0. (DS. (3− 3)+( 3−1) ln 2) 310. (x2 + y 2)dS, (Σ) l` phˆn m˘t paraboloid x2 + y 2 = 2z du.o.c a ` a a . . (Σ) √ ´ ’c˘t ra bo a .i m˘t ph˘ng z = 1. (DS. 55 + 9 3 ) a ’ a . 6511. 1 + 4x2 + 4y 2dS, (Σ) l` phˆn m˘t paraboloid z = 1−x2 −y 2 a ` a a . (Σ)gi´.i han bo.i c´c m˘t ph˘ng z = 0 v` z = 1. (DS. 3π) o . ’ a a . ’ a a12. (x2 + y 2)dS, (Σ) l` phˆn m˘t n´n z = a ` a a o . x2 + y 2 n˘m gi˜.a ` a u (Σ) √ π 2 a a . ’c´c m˘t ph˘ng z = 0 v` z = 1. (DS. a a ) 213. a ` (xy + yz + zx)dS, (Σ) l` phˆn m˘t n´n z = a a o . ` x2 + y 2 n˘m a (Σ) √ 64a4 2trong m˘t tru x2 + y 2 = 2ax (a > 0). (DS. a . . ) 1514. (x2 + y 2 + z 2)dS, (Σ) l` mat cˆu. (DS. 4π) a . ` a (Σ)15. xds, (Σ) l` phˆn m˘t du.o.c c˘t ra t`. partaboloid 10x = y 2 +z 2 a ` a a . . a ´ u (Σ)
  • 171. 170 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 50π √ bo.i m˘t ph˘ng x = 10. (DS. ’ a . ’ a (1 + 25 5)) 3 Su. dung cˆng th´.c t´nh diˆn t´ m˘t S(Σ) = ’ . o u ı e ıch a . . ’ dS dˆ t´nh diˆn e ı e . (Σ) ıch ’ ` t´ cua phˆn m˘t (Σ) nˆu a a . e´ a ` . ’ a ` 16. (Σ) l` phˆn m˘t ph˘ng 2x + 2y + z = 8a n˘m trong m˘t tru a a a a . . 2 2 2 2 x + y = R . (DS. 3πR ) 17. (Σ) l` phˆn m˘t tru y + z 2 = R2 n˘m trong m˘t tru a ` a a . . ` a a . . 2 2 2 2 x + y = R . (DS. 8R ) 18. (Σ) l` phˆn m˘t paraboloid x2 + y 2 = 6z n˘m trong m˘t tru a ` a a . ` a a . . 2 2 x + y = 27. (DS. 42π) 19. (Σ) l` phˆn m˘t cˆu x2 + y 2 + z 2 = 3a2 n˘m trong paraboloid a ` a a ` . a ` a 2 2 2 √ x + y = 2az. (DS. 2πa (3 − 3)) 20. (Σ) l` phˆn m˘t n´n z 2 = 2xy n˘m trong g´c phˆn t´m I gi˜.a a ` a a o . ` a o ` a a u hai m˘t ph˘ a . a’ ng x = 2, y = 4. (DS. 16) 21. (Σ) l` phˆn m˘t tru x2 + y 2 = Rx n˘m trong m˘t cˆu a ` a a . . ` a a ` . a 2 2 2 2 2 x + y + z = R . (DS. 4R ) T´ c´c t´ phˆn m˘t theo toa dˆ sau: ınh a ıch a a . . o . 22. a ı a ` dxdy, (Σ) l` ph´a ngo`i phˆn m˘t n´n z = a a o . x2 + y 2 khi (Σ) 0 z 1. (DS. −π) 23. a ıa e ’ ` ’ ydzdx, (Σ) l` ph´ trˆn cua phˆn m˘t ph˘ng x + y + z = a a a . a (Σ) a3 ` ` a (a > 0) n˘m trong g´c phˆn t´m I. (DS. a o a ) 6 24. a ıa e ’ ` ’ xdydz + ydzdx + zdxdy, (Σ) l` ph´ trˆn cua phˆn m˘t ph˘ng a a . a (Σ) x + z − 1 = 0 n˘m gi˜.a hai m˘t ph˘ng y = 0 v` y = 4 v` thuˆc v`o ` a u a . ’ a a a o a . o ` n t´m I. (DS. 4) g´c phˆ a a
  • 172. 12.4. T´ phˆn m˘t ıch a a . 17125. a ı e ’ ` − xdydz + zdzdx + 5dxdy, (Σ) l` ph´a trˆn cua phˆn m˘t a a . (Σ) ’ a o o . ` aph˘ng 2x + 3y + z = 6 thuˆc g´c phˆn t´m I. (DS. 6) a26. a ıa e ’ yzdydz + xzdxdz + xydxdy, (Σ) l` ph´ trˆn cua tam gi´c tao a . (Σ)bo.i giao tuyˆn cua m˘t ph˘ng x + y + z = a v´.i c´c m˘t ph˘ng toa ’ ´ e ’ a . ’ a o a a . ’ a . a4dˆ. (DS. ) o . 827. x2dydz + z 2dxdy, (Σ) l` ph´a ngo`i cua phˆn m˘t n´n a ı a ’ ` a a o . (Σ) 4 x2 + y 2 = z 2 , 0 z 1. (DS. − ) 328. a ı a ` a a ` xdydz + ydzdx + zdxdy, (Σ) l` ph´a ngo`i phˆn m˘t cˆu . a (Σ) x + y 2 + z 2 = a2. 2 (DS. 4πa3)29. x2dydz − y 2 dzdx + z 2 dxdy, (Σ) l` ph´a ngo`i cua m˘t cˆu a ı a ’ a ` . a (σ) πa4 x2 + y 2 + z 2 = R2 thuˆc g´c phˆn t´m I. (DS. o o . ` a a ) 830. 2dxdy + ydzdx − x2 zdydz, (Σ) l` ph´ ngo`i cua phˆn m˘t a ıa a ’ ` a a . (Σ) 4π 4elipxoid 4x2 + y 2 + 4z 2 = 4 thuˆc g´c phˆn t´m I. (DS. o o . ` a a − ) 3 1531. (y 2 + z 2 )dxdy, (Σ) l` ph´a ngo`i cua m˘t tru z 2 = 1 − x2, a ı a ’ a . . (Σ) π 0 y 1. (DS. ) 332. (z − R)2 dxdy, (Σ) l` ph´a ngo`i cua nu.a m˘t cˆu a ı a ’ ’ a ` . a (Σ) 5π x2 + y 2 + (z − R)2 = R2 , R z 2R. (DS. − ) 24
  • 173. 172 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 33. x2dydz + y 2 dzdx + z 2 dxdy, (Σ) l` ph´a ngo`i cua phˆn m˘t a ı a ’ ` a a . (Σ) 3πa4 cˆu x2 + y 2 + z 2 = a2 thuˆc g´c phˆn t´m I. (DS. ` a o o . ` a a ) 8 34. z 2dxdy, (σ) l` ph´ trong cua m˘t elipxoid a ıa ’ a . (Σ) x2 + y 2 + 2z 2 = 2. (DS. 0) 35. a ’ a ` (z + 1)dxdy, (Σ) l` ph´a ngo`i cua m˘t cˆu a ı . a (Σ) 2 2 2 2 4πR3 x + y + z = R . (DS. ) 3 36. x2dydz + y 2 dzdx + z 2 dxdy, (Σ) l` ph´a ngo`i cua m˘t cˆu a ı a ’ a ` . a (Σ) 8πR3 (x − a)2 + (y − b)2 + (z − c)2 = R2 . (DS. (a + b + c)) 3 37. x2y 2zdxdy, (Σ) l` ph´a trong cua nu.a du.´.i m˘t cˆu a ı ’ ’ o a `. a (Σ) 2πR7 x2 + y 2 + z 2 = R2 . (DS. ) 105 38. xzdxdy + xydydz + yzdxdz, (Σ) l` ph´ ngo`i cua t´. diˆn tao a ıa a ’ u e . . (Σ) 1 bo.i c´c m˘t ph˘ng toa dˆ v` m˘t ph˘ng x + y + z = 1. (DS. ) ’ a a. ’ a . o a a . . ’ a 8 Chı dˆ n. Su. dung nhˆn x´t nˆu trong phˆn l´ thuyˆt. ˜ ’ a ’ . a e e . ` y a ´ e 39. a ı a ’ yzdydz + xzdxdz + xydxdy, (Σ) l` ph´a ngo`i cua m˘t biˆn a . e (Σ) t´. diˆn lˆp bo.i c´c m˘t ph˘ng x = 0, y = 0, z = 0, x + y + z = a. u e a. . ’ a a . ’ a (DS. 0) 40. x2dydz + y 2 dzdx + z 2 dxdy, (Σ) l` ph´a ngo`i cua nu.a trˆn a ı a ’ ’ e (Σ)
  • 174. 12.4. T´ phˆn m˘t ıch a a . 173 πR4m˘t cˆu x2 + y 2 + z 2 = R2 (z a ` . a 0). (DS. ) 2 Ap dung cˆng th´.c Gauss-Ostrogradski dˆ t´nh t´ phˆn m˘t theo ´ . o u ’ e ı ıch a a .ph´ ngo`i cua m˘t (Σ) (nˆu m˘t khˆng k´ th` bˆ sung dˆ n´ tro. th`nh ıa a ’ a . e´ a . o ın ı o ’ ’ o ’ a ek´ ın)41. x2dydz + y 2dzdx + z 2 dxdy, (Σ) l` m˘t cˆu a a ` . a (Σ) 8π (x − a)2 + (y − b)2 + (z − c)2 = R2 . (DS. (a + b + c)R3 ) 342. xdydz + ydzdx + zdxdy, (Σ) l` m˘t cˆu x2 + y 2 + z 2 = R2 . a a ` . a (Σ) (DS. 4πR3 )43. 4x3 dydz + 4y 3 dzdx − 6z 2 dxdy, (Σ) l` biˆn cua phˆn h`nh a e ’ ` a ı (Σ) tru x2 + y 2 . a2 , 0 z h. (DS. 6πa2(a2 − h2 ))44. a ` (y − z)dydz + (z − x)dzdx + (x − y)dxdy, (Σ) l` phˆn m˘t a a . (σ) n´n x2 + y 2 = z 2 , 0 o x h. (DS. 0) ’ a˜ ı o ın e ` o a ’ ` a a . ’ Chı dˆ n. V` (Σ) khˆng k´ nˆn cˆn bˆ sung phˆn m˘t ph˘ng z = h an˘m trong n´n dˆ thu du.o.c m˘t k´ ` a o e ’ . a ın. .45. a e ’ ` dydz + zxdzdx + xydxdy, (Σ) l` biˆn cua miˆn e (Σ) {(x, y, z) : x2 + y 2 a2, 0 z h}. (DS. 0)46. ydydz + zdzdx + xdxdy, (Σ) l` m˘t cua h`nh ch´p gi´.i han a a ’ . ı o o . (Σ)bo.i c´c m˘t ph˘ng ’ a a. ’ a x + y + z = a (a > 0), x = 0, y = 0, z = 0. (DS. 0)47. x3dydz + y 3dzdx + z 3 dxdy, (Σ) l` m˘t cˆu x2 + y 2 + z 2 = x. a a ` . a (Σ)
  • 175. 174 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e π (DS. ) 5 48. x3dydz + y 3dzdx + z 3dxdy, (Σ) l` m˘t cˆu x2 + y 2 + z 2 = a2. a a ` . a (Σ) 12πa5 (DS. ) 5 x2 y 2 z 2 49. z 2dxdy, (Σ) l` m˘t elipxoid 2 + 2 + 2 = 1. (DS. 0) a a. a b c (Σ) ’ a˜ Chı dˆ n. Xem v´ du 10, muc III. ı . . x2 y 2 z 2 50. xdydz + ydzdx + zdxdy, (Σ) l` m˘t elipxoid 2 + 2 + 2 = 1. a a. a b c (Σ) (Ds. 4πabc) 51. xdydz + ydzdx + zdxdy, (Σ) l` biˆn h` tru x2 + y 2 a e ınh . a2, (Σ) −h z h. (DS. 6πa2h) 52. x2dydz + y 2 dzdx + z 2 dxdy, (Σ) l` biˆn cua h`nh lˆp phu.o.ng a e ’ ı a . (Σ) 0 x a, 0 y a, 0 z a. (DS. 3a4 ) Dˆ ´p dung cˆng th´.c Stokes, ta lu.u y lai quy u.´.c ’ ea . o u ´ . o Hu o.´.ng du.o.ng cua chu tuyˆn ∂Σ cua m˘t (Σ) du.o.c quy u.´.c nhu. ’ ´ e ’ a o . . sau: Nˆu mˆt ngu.`.i quan tr˘c d´.ng trˆn ph´a du.o.c chon cua m˘t (t´.c ´ o e . o ´ u a e ı . . ’ a u . a .´.ng t`. chˆn dˆn dˆu tr`ng v´.i hu.´.ng cua vecto. ph´p tuyˆn) th` l` hu o u a e ` ´ a u o o ’ a ´ e ı khi ngu o .`.i quan s´t di chuyˆn trˆn ∂Σ theo hu.´.ng d´ th` m˘t (Σ) luˆn a ’ e e o o ı a o . o ` luˆn n˘m bˆn tr´i. a e a Ap dung cˆng th´.c Stokes dˆ t´nh c´c t´ch phˆn sau ´ . o u ’ e ı a ı a 53. ´ e ’ ’ xydx + yzdy + xzdz, C l` giao tuyˆn cua m˘t ph˘ng 2x − 3y + a a . a C 4z − 12 = 0 v´.i c´c m˘t ph˘ng toa dˆ. (DS. −7) o a a . ’ a . o .
  • 176. 12.4. T´ phˆn m˘t ıch a a . 17554. ydx+zdy +xdz, C l` du.`.ng tr`n x2 +y 2 +z 2 = R2 , x+y +z = 0 a o o Cc´ hu.´.ng ngu.o.c chiˆu kim dˆng hˆ nˆu nh`n t`. phˆn du.o.ng truc Ox. o o . ` e ` o o ´ ` e ı u ` a . √ 2(DS. − 3πR )55. (y − z)dx + (z − x)dy + (x − y)dz, C l` elip x2 + y 2 = a2, a Cx z + = 1 (a > 0, h > 0) c´ hu.´.ng ngu.o.c chiˆu kim dˆng hˆ nˆu o o . ` e ` o o ´ ` ea hnh` t`. diˆm (2a, 0, 0). (DS. −2πa(a + h)) ın u e ’56. (y−z)dx+(z−x)dy+(x−y)dz, C l` du.`.ng tr`n x2 +y 2 +z 2 = a2, a o o C πy = xtgα, 0 < α < c´ hu.´.ng ngu.o.c chiˆu kim dˆng hˆ nh`n t`. o o . ` e ` o ` o ı u 2 √ π e’ diˆm (2a, 0, 0). (DS. 2 2πa2 sin − α)) 457. (y − z)dx + (z − x)dy + (x −y)dz, C l` elip x2 + y 2 = 1, x + z = 1 a Cc´ hu.´.ng ngu.o.c chiˆu kim dˆng hˆ nˆu nh`n t`. phˆn du.o.ng truc Oz. o o . ` e ` o o ´ ` e ı u ` a .(DS. −4π)58. (y 2 − z 2)dx + (z 2 − x2)dy + (x2 − y 2)dz, C l` biˆn cua thiˆt diˆn a e ’ ´ e e . Ccua lˆp phu.o.ng 0 ’ a . x a, 0 y a, 0 z a v´.i m˘t ph˘ng o a . ’ a 3ax+y+z = c´ hu.´.ng ngu.o.c chiˆu kim dˆng hˆ nˆu nh`n t`. diˆm o o . ` e ` o o ´ ` e ı u e ’ 2 9(2a, 0, 0). (DS. − a3 ) 259. exdx + z(x2 + y 2)3/2dy + yz 3dz, C l` giao tuyˆn cua m˘t z = a ´ ’ e a . C x2 + y 2 v´.i c´c m˘t ph˘ng x = 0, x = 2, y = 0, y = 1. o a a . ’ a (DS. −14)60. 8y (1 − x2 − z 2 )3 dx + xy 3dy + sin zdz, C l` biˆn cua mˆt phˆn a e ’ o . ` a Ctu. elipxoid 4x2 + y 2 + 4z 2 = 4 n˘m trong g´c phˆn t´m th´. I. ` a o ` a a u
  • 177. 176 Chu.o.ng 12. T´ phˆn h`m nhiˆu biˆn ıch a a ` e ´ e 32 (DS. ) 5
  • 178. Chu.o.ng 13 y ´ e ˜L´ thuyˆt chuˆ i o 13.1 Chuˆ i sˆ du.o.ng . . . . . . . . . . . . . . . . 178 ˜ o o ´ 13.1.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 178 a . ıa ’ 13.1.2 Chuˆ i sˆ du.o.ng . . . . . . . . . . . . . . . . 179 ˜ ´ o o ˜ o .. . ´ 13.2 Chuˆ i hˆi tu tuyˆt d ˆi v` hˆi tu khˆng o e o a o . . o e . ´ tuyˆt d ˆi . . . . . . . . . . . . . . . . . . . . 191 o 13.2.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 191 a . ıa ’ ˜ o ´ a a a ´ 13.2.2 Chuˆ i dan dˆu v` dˆu hiˆu Leibnitz . . . . 192 e . 13.3 Chuˆ i l˜y th`.a . . . . . . . . . . . . . . . . . 199 ˜ u o u 13.3.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 199 a . ıa ’ 13.3.2 Diˆu kiˆn khai triˆn v` phu.o.ng ph´p khai - `e e . ’ e a a triˆ e’n . . . . . . . . . . . . . . . . . . . . . . 201 ˜ 13.4 Chuˆ i Fourier . . . . . . . . . . . . . . . . . . 211 o 13.4.1 C´c dinh ngh˜ co. ban . . . . . . . . . . . . 211 a . ıa ’ 13.4.2 Dˆu hiˆu du vˆ su. hˆi tu cua chuˆ i Fourier 212 ´ a e . ’ ` . o . ’ e . ˜ o
  • 179. 178 Chu.o.ng 13. L´ thuyˆt chuˆ i y ´ e ˜ o 13.1 Chuˆ i sˆ du.o.ng ˜ o o ´ 13.1.1 C´c dinh ngh˜ co. ban a . ıa ’ Gia su. cho d˜y sˆ (an ). Biˆu th´.c dang ’ ’ a o ´ ’ e u . ∞ a1 + a2 + · · · + an + · · · = an = an (13.1) n=1 n 1 du.o.c goi l` chuˆ i sˆ (hay do.n gian l` chuˆ i). C´c sˆ a1, . . . , an , . . . . . a ˜ o o ´ ’ a ˜ o a o ´ du.o.c goi l` c´c sˆ hang cua chuˆ i, sˆ hang an goi l` sˆ hang tˆng qu´t . . a a o . ´ ’ ˜ ´ o o . . a o . ´ o’ a ’ ˜ o’ ´ o . ` a e ’ ˜ cua chuˆ i. Tˆng n sˆ hang dˆu tiˆn cua chuˆ i du . o o .o.c goi l` tˆng riˆng ’ . a o e th´u. n cua chuˆ i v` k´ hiˆu l` sn , t´.c l` ’ ˜ o a y e a u a . sn = a1 + a2 + · · · + an . ´ ´ ’ ˜ o a o . e a o ’ e ’ ˜ . V` sˆ sˆ hang cua chuˆ i l` vˆ han nˆn c´c tˆng riˆng cua chuˆ i lˆp ı o o . o a a a o . a o ’ th`nh d˜y vˆ han c´c tˆng riˆng s1 , s2 , . . . , sn , . . . e Dinh ngh˜ 13.1.1. Chuˆ i (13.1) du.o.c goi l` chuˆ i hˆi tu nˆu d˜y -. ıa ˜ o . . a ˜ o . e a o . ´ ’ e ’ o o o .i han h˜.u han v` gi´.i han d´ du.o.c c´c tˆng riˆng (sn ) cua n´ c´ gi´ . a o u a o . o . . goi l` tˆ . a o ’ng cua chuˆ i hˆi tu. Nˆu d˜y (sn ) khˆng c´ gi´.i han h˜.u han ’ ˜ o . o . ´ a e o o o . u . ı ˜ th` chuˆ i (13.1) phˆn k`. o a y -i ` e ` e ’ ˜ ´ ’ D.nh l´ 13.1.1. Diˆu kiˆn cˆn dˆ chuˆ i (13.1) hˆi tu l` sˆ hang tˆng y e . a o o . a o . . o a ’ o ` ´ qu´t cua n´ dˆn dˆn 0 khi n → ∞, t´ a a e u.c l` lim an = 0. n→∞ Dinh l´ 13.1.1 chı l` diˆu kiˆn cˆn ch´. khˆng l` diˆu kiˆn du. . y ’ a ` e e ` . a u o a `e e . ’ Nhu.ng t`. d´ c´ thˆ r´t ra diˆu kiˆn du dˆ chuˆ i phˆn k`: Nˆu u o o e u ’ ` e e . ’ e ’ ˜ o a y e´ ˜ lim an = 0 th` chuˆ i ı o an phˆn k`. a y n→∞ n 1 ˜ Chuˆ i o an thu du.o.c t`. chuˆ i . u ˜ o ´ ’ ´ an sau khi c˘t bo m sˆ hang a o . n m+1 n 1 dˆu tiˆn du.o.c goi l` phˆn du. th´. m cua chuˆ i ` a e . . a ` a u ’ ˜ o ´ ˜ an . Nˆu chuˆ i (13.1) e o n 1 hˆi tu th` moi phˆn du. cua n´ dˆu hˆi tu, v` mˆt phˆn du. n`o d´ o . ı . . `a ’ o ` e o . a o . . `a a o hˆi tu th` ban thˆn chuˆ i c˜ng hˆi tu. Nˆu phˆn du. th´. m cua chuˆ i o . ı ’ . a ˜ u o o . . ´ e ` a u ’ ˜ o
  • 180. 13.1. Chuˆ i sˆ du.o.ng ˜ ´ o o 179 ’ ’ o a ` ˜ o .(13.1) hˆi tu v` tˆng cua n´ b˘ng Rm th` s = sm + Rm . Chuˆ i hˆi tu o . a o . ı o . o a ınh a ´c´ c´c t´ chˆt quan trong l`. a (i) V´.i sˆ m cˆ dinh bˆt k` chuˆ i (13.1) v` chuˆ i phˆn du. th´. m o o ´ ´ o . ´ a y ˜ o a ˜ o ` a u ’ o ` o .i hˆi tu ho˘c dˆng th`.i phˆn k`.cua n´ dˆng th` o . a ` o . . o o a y ´ e ˜ (ii) Nˆu chuˆ i (13.1) hˆi tu th` Rm → 0 khi m → ∞ o o . ı . ´ (iii) Nˆu c´c chuˆ i e a ˜ o an v`a ` ´ bn hˆi tu v` α, β l` h˘ng sˆ th` o . a . a a o ı n 1 n 1 (αan + βbn ) = α an + β bn . n 1 n 1 n 113.1.2 Chuˆ i sˆ du.o.ng ˜ o o ´ ˜ o o ´Chuˆ i sˆ an du.o.c goi l` chuˆ i sˆ du.o.ng nˆu an 0 ∀ n ∈ N. Nˆu . . a ˜ o o ´ e´ e´ n 1an > 0 ∀ n th` chuˆ i du.o.c goi l` chuˆ i sˆ du.o.ng thu.c su.. ı ˜ o . . a ˜ o o ´ . . ’ . ˜ ´ Tiˆu chuˆn hˆi tu. Chuˆ i sˆ du e a o . o o .o.ng hˆi tu khi v` chı khi d˜y tˆng o . a ’ a o ’ . e ’ o . ariˆng cua n´ bi ch˘n trˆn. . e Nh`. diˆu kiˆn n`y, ta c´ thˆ thu du.o.c nh˜.ng dˆu hiˆu du sau dˆy: o ` e e a . o e ’ . u ´ a e ’ . a a´ e a ’ ’ Dˆu hiˆu so s´nh I. Gia su . cho hai chuˆ i sˆ ˜ o o ´ . A: an , an 0 ∀ n ∈ N v` B : a bn , bn 0 ∀n ∈ N n 1 n 1v` an bn ∀ n ∈ N. Khi d´: a o ´ e ˜ ´ o o o . ı . ˜ o o ´ (i) Nˆu chuˆ i sˆ B hˆi tu th` chuˆ i sˆ A hˆi tu, o . . (ii) Nˆ e ˜ o o a y ı ˜ o ´u chuˆ i sˆ A phˆn k` th` chuˆ i sˆ B phˆn k`. ´ o ´ a y a´ Dˆu hiˆu so s´nh II. Gia su a e a ’ ’ . c´c chuˆ i sˆ A v` B l` nh˜.ng chuˆ i ˜ ´ o o a a u ˜ o . ´sˆ du o .o.ng thu.c su. v` ∃ lim an = λ (r˜ r`ng l` 0 λ . . a n→∞ b o a a +∞). Khi nd´: o (i) Nˆu λ < ∞ th` t`. su. hˆi tu cua chuˆ i sˆ B k´o theo su. hˆi tu ´ e ı u . o . ’ . ˜ o o ´ e . o . . ’ ˜ ´cua chuˆ i sˆ A o o (ii) Nˆu λ > 0 th` t`. su. hˆi tu cua chuˆ i sˆ A k´o theo su. hˆi tu e´ ı u . o . ’ . ˜ ´ o o e . o . . ’cua chuˆ o˜ i sˆ B o ´
  • 181. 180 Chu.o.ng 13. L´ thuyˆt chuˆ i y ´ e ˜ o (iii) Nˆu 0 < λ < +∞ th` hai chuˆ i A v` B dˆng th`.i hˆi tu ho˘c ´ e ı ˜ o a ` o o o . a . . ` dˆng th` o o.i phˆn k`. a y Trong thu.c h`nh dˆu hiˆu so s´nh thu.`.ng du.o.c su. dung du.´.i dang . a ´ a e . a o . ’ . o . “ thu a.c h`nh” sau dˆy: a . ´ Dˆu hiˆu thu a a e .c h`nh. Nˆu dˆi v´.i d˜y sˆ du.o.ng (an ) tˆn tai c´c sˆ ´ ´ e o o a o ´ ` . a o o ´ . . C p v` C > 0 sao cho an ∼ p , n → ∞ th` chuˆ i a ı ˜ o ´ an hˆi tu nˆu p > 1 o . e . n n 1 v` phˆn k` nˆu p 1. a a y e ´ C´c chuˆ i thu.`.ng du.o.c d`ng dˆ so s´nh l` a ˜ o o . u ’ e a a ˜ ´ ´ 1) Chuˆ i cˆp sˆ nhˆn o a o a n aq , a = 0 hˆi tu khi 0 q < 1 v` phˆn o . . a a n 0 k` khi q y 1. 1 ˜ 2) Chuˆ i Dirichlet: o hˆi tu khi α > 1 v` phˆn k` khi α o . . a a y 1. α n 1 n 1 ˜ Chuˆ i phˆn k` o a y ˜ ` o goi l` chuˆ i diˆu h`a. . a o e n 1 n T`. dˆu hiˆu so s´nh I v` chuˆ i so s´nh 1) ta r´t ra: u a ´ e . a a ˜ o a u ´ a e . ´ Dˆu hiˆu D’Alembert. Nˆu chuˆ e ˜ i a1 + a2 + · · · + an + . . . , an > 0 o ∀ n c´ o an+1 lim =D n→∞ an ı ˜ o . th` chuˆ i hˆi tu khi D < 1 v` phˆn k` khi D > 1. o . a a y ´ e. ´ e ˜ Dˆu hiˆu Cauchy. Nˆu chuˆ i a1 + a2 + · · · + an + . . . , an a o 0 ∀n c´ o √ lim n an = C n→∞ ı ˜ o . th` chuˆ i hˆi tu khi C < 1 v` phˆn k` khi C > 1. o . a a y Trong tru.`.ng ho.p khi D = C = 1 th` ca hai dˆu hiˆu n`y dˆu o . ı ’ ´ a e . a ` e khˆng cho cˆu tra l` o a ’ o .i kh˘ng dinh v` tˆn tai chuˆ i hˆi tu lˆ n chuˆ i ’ a ı ` o . ˜ o . a o . ˜ ˜ o . phˆn k` v´ a y o .i D ho˘c C b˘ng 1. a ` a . ´ . ´ Dˆu hiˆu t´ phˆn. Nˆu h`m f(x) x´c dinh ∀ x a e ıch a e a a . 1 khˆng ˆm o a a ’ v` giam th` chuˆ i ı ˜ o a ’ f(n) hˆi tu khi v` chı khi t´ phˆn suy rˆng o . . ıch a o . n 1
  • 182. 13.1. Chuˆ i sˆ du.o.ng ˜ ´ o o 181∞ f (x)dx hˆi tu. o . .0 1 T`. dˆu hiˆu t´ch phˆn suy ra chuˆ i u a ´ e ı . a ˜ o α hˆi tu khi α > 1 v` o . . a n 1 n 1 e´phˆn k` khi 0 < α 1. Nˆu α 0 th` do an = α → 0 khi α 0 v` a y ı a n ˜ an → ∞ nˆn chuˆ i d˜ cho c˜ng phˆn k`. e o u a y CAC V´ DU ´ I .V´ du 1. Khao s´t su. hˆi tu cua c´c chuˆ i ı . ’ a . o . ’ a . ˜ o 1 1 1) ; 2) · n 1 n(n + 1) n 7 nln n Giai. 1) Su. dung bˆt d˘ng th´.c hiˆn nhiˆn ’ ’ . ´ ’ a a u ’ e e 1 1 > · n(n + 1) n+1 1 ˜ V` chuˆ i ı o l` phˆn du. sau sˆ hang th´. nhˆt cua chuˆ i diˆu a ` a ´ o . u a ’´ ˜ e o ` n 1n+1h`a nˆn n´ phˆn k`. o e o a y ´ e . a ˜ Do d´ theo dˆu hiˆu so s´nh I chuˆ i d˜ cho phˆn k`. o a o a a y