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# Bai tap-toan-cao-cap-tap-3 nt-thanh[vnmath.com]

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### Bai tap-toan-cao-cap-tap-3 nt-thanh[vnmath.com]

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 .