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Table of Integrals∗                             Basic Forms                                                          Integ...
Integrals with Logarithms                               1              x2 ± a2 dx =       x x2 ± a2                       ...
x2  2x   2                                                                1       x2 eax dx =            − 2 + 3      eax ...
Products of Trigonometric Functions and                     2        1                                                    ...
1                                                            1 cos ax sinh bxdx =          [b cos ax cosh bx+             ...
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Integral table

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Integral table

  1. 1. Table of Integrals∗ Basic Forms Integrals with Roots 1 √ 2 xn dx = xn+1 + c (1) x − adx = (x − a)3/2 + C (17) n+1 3 1 1 √ dx = ln x + c (2) √ dx = 2 x ± a + C (18) x x±a 1 √ udv = uv − vdu (3) √ dx = 2 a − x + C (19) a−x 1 1 dx = ln |ax + b| + c (4) ax + b a √ 2 2 x x − adx = a(x − a)3/2 + (x − a)5/2 + C (20) 3 5 Integrals of Rational Functions √ 2b 2x √ 1 1 ax + bdx = + ax + b + C (21) dx = − +c (5) 3a 3 (x + a)2 x+a 2 (x + a)n+1 (ax + b)3/2 dx = (ax + b)5/2 + C (22) n (x + a) dx = + c, n = −1 (6) 5a n+1 x 2 √ √ dx = (x ± 2a) x ± a + C (23) x±a 3 (x + a)n+1 ((n + 1)x − a) x(x + a)n dx = +c (7) (n + 1)(n + 2) x 1 dx = − x(a − x) dx = tan−1 x + c (8) a−x 1 + x2 x(a − x) − a tan−1 +C (24) 1 1 x x−a dx = tan−1 + c (9) a2 + x2 a a x 1 x dx = ln |a2 + x2 | + c (10) dx = x(a + x) a2 +x 2 2 a+x √ √ x2 x − a ln x + x + a + C (25) dx = x − a tan−1 + c (11) a2 +x 2 a x3 1 1 √ dx = x2 − a2 ln |a2 + x2 | + c (12) a2 + x2 2 2 x ax + bdx = 2 √ 1 2 2ax + b 2 (−2b2 + abx + 3a2 x2 ) ax + b + C (26) dx = √ tan−1 √ + C (13) 15a ax 2 + bx + c 2 4ac − b 4ac − b2 1 1 a+x 1 dx = ln , a=b (14) x(ax + b)dx = (2ax + b) ax(ax + b) (x + a)(x + b) b−a b+x 4a3/2 √ x a −b2 ln a x + a(ax + b) + C (27) 2 dx = + ln |a + x| + C (15) (x + a) a+x x 1 b b2 x dx = ln |ax2 + bx + c| x3 (ax + b)dx = − 2 + x3 (ax + b) ax2 + bx + c 2a 12a 8a x 3 b 2ax + b b3 √ − √ tan−1 √ + C (16) + 5/2 ln a x + a(ax + b) + C (28) a 4ac − b 2 4ac − b2 8a ∗ c 2007. From http://integral-table.com, last revised December 6, 2007. This material is provided as is without warranty or representationabout the accuracy, correctness or suitability of this material for any purpose. Some restrictions on use and distribution may apply, including theterms of the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. See the web site for details. The formula numberson this document may be different from the formula numbers on the web page. 1
  2. 2. Integrals with Logarithms 1 x2 ± a2 dx = x x2 ± a2 2 ln axdx = x ln ax − x + C (41) 1 ± a2 ln x + x2 ± a2 + C (29) 2 ln ax 1 2 dx = (ln ax) + C (42) x 2 1 a2 − x2 dx = x a2 − x2 b 2 ln(ax + b)dx = x+ ln(ax + b) − x + C, a = 0 (43) 1 x a + a2 tan−1 √ +C (30) 2 a2 − x2 1 2 3/2 ln a2 x2 ± b2 dx = x ln a2 x2 ± b2 x x2 ± a2 dx = x ± a2 +C (31) 3 2b ax + tan−1 − 2x + C (44) 1 a b √ dx = ln x + x2 ± a2 + C (32) x 2 ± a2 1 x ln a2 − b2 x2 dx = x ln ar − b2 x2 √ dx = sin−1 + C (33) a2 − x2 a 2a bx + tan−1 − 2x + C (45) x b a √ dx = x2 ± a2 + C (34) x2 ± a2 x 1 2ax + b √ dx = − a2 − x2 + C (35) ln ax2 + bx + c dx = 4ac − b2 tan−1 √ a2 − x2 a 4ac − b2 b − 2x + + x ln ax2 + bx + c + C (46) 2a x2 1 √ dx = x x2 ± a2 x 2 ± a2 2 1 2 bx 1 2 a ln x + x2 ± a2 + C (36) x ln(ax + b)dx = − x 2 2a 4 1 b2 + x2 − 2 ln(ax + b) + C (47) 2 a b + 2ax ax2 + bx + cdx = ax2 + bx + c 4a 1 4ac − b2 x ln a2 − b2 x2 dx = − x2 ++ ln 2ax + b + 2 a(ax2 + bx+ c) + C (37) 2 8a3/2 1 a2 x2 − 2 ln a2 − b2 x2 + C (48) 2 b 1 √x ax2 + bx + c = 5/2 2 a ax2 + bx + c Integrals with Exponentials 48a− 3b2 + 2abx + 8a(c + ax2 ) √ 1 ax eax dx = e +C (49)+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + x (38) a √ √ ax 1 √ ax i π √ 1 xe dx = xe + 3/2 erf i ax + C, √ dx = a 2a x ax2 + bx + c 2 2 1 where erf(x) = √ e−t dtet (50) √ ln 2ax + b + 2 a(ax2 + bx + c) + C (39) π 0 a xex dx = (x − 1)ex + C (51) x 1 √ dx = ax2 + bx + c x 1 ax2 + bx + c a xeax dx = − eax + C (52) b a a2 + 3/2 ln 2ax + b + 2 a(ax2 + bx + c) + C (40) 2a x2 ex dx = x2 − 2x + 2 ex + C (53) 2
  3. 3. x2 2x 2 1 x2 eax dx = − 2 + 3 eax + C (54) sin2 x cos xdx = sin3 x + C (68) a a a 3 x3 ex dx = x3 − 3x2 + 6x − 6 ex + C (55) cos[(2a − b)x] cos bx cos2 ax sin bxdx = − 4(2a − b) 2b cos[(2a + b)x] 1 − +C (69) xn eax dx = (−1)n Γ[1 + n, −ax], 4(2a + b) a ∞ where Γ(a, x) = ta−1 e−t dt (56) 1 cos2 ax sin axdx = − cos3 ax + C (70) x 3a √ 2 i π √ eax dx = − √ erf ix a (57) x sin 2ax sin[2(a − b)x] 2 a sin2 ax cos2 bxdx = − − 4 8a 16(a − b)Integrals with Trigonometric Functions sin 2bx sin[2(a + b)x] + − +C (71) 8b 16(a + b) 1 sin axdx = − cos ax + C (58) x sin 4ax a sin2 ax cos2 axdx = − +C (72) 8 32a x sin 2ax sin2 axdx = − +C (59) 1 2 4a tan axdx = − ln cos ax + C (73) a 1 n tan2 axdx = −x + tan ax + C (74) sin axdx = a 1 1 1−n 3 − cos ax 2 F1 , , , cos2 ax + C (60) tann+1 ax a 2 2 2 tann axdx = × a(1 + n) 3 cos ax cos 3ax n+1 n+3 sin3 axdx = − + +C (61) 2 F1 , 1, , − tan2 ax + C (75) 4a 12a 2 2 1 1 1 cos axdx = sin ax + C (62) tan3 axdx = ln cos ax + sec2 ax + C (76) a a 2a x sin 2ax cos2 axdx = + +C (63) 2 4a sec xdx = ln | sec x + tan x| + C x 1 = 2 tanh−1 tan +C (77)cosp axdx = − cos1+p ax× 2 a(1 + p) 1+p 1 3+p 1 2 F1 , , , cos2 ax + C (64) sec2 axdx = tan ax + C (78) 2 2 2 a 3 sin ax sin 3ax cos3 axdx = + +C (65) 1 1 4a 12a sec3 xdx = sec x tan x + ln | sec x tan x| + C (79) 2 2 cos[(a − b)x] cos ax sin bxdx = − sec x tan xdx = sec x + C (80) 2(a − b) cos[(a + b)x] + C, a = b (66) 1 2(a + b) sec2 x tan xdx = sec2 x + C (81) 2 1 2 sin[(2a − b)x] secn x tan xdx = secn x + C, n = 0 (82)sin ax cos bxdx = − n 4(2a − b) sin bx sin[(2a + b)x] + − +C (67) x 2b 4(2a + b) csc xdx = ln tan + C = ln | csc x − cot x| + C (83) 2 3
  4. 4. Products of Trigonometric Functions and 2 1 Exponentials csc axdx = − cot ax + C (84) a 1 x ex sin xdx = e (sin x − cos x) + C (99) 1 1 2 csc3 xdx = − cot x csc x + ln | csc x − cot x| + C (85) 2 2 1 ebx sin axdx = ebx (b sin ax − a cos bx) + C (100) 1 a2 + b2 cscn x cot xdx = − cscn x + C, n = 0 (86) n 1 x ex cos xdx = e (sin x + cos x) + C (101) 2 sec x csc xdx = ln | tan x| + C (87) 1 ebx cos axdx = ebx (a sin ax + b cos ax) + C (102)Products of Trigonometric Functions and Monomials a2 + b2 1 x xex sin xdx = e (cos x − x cos x + x sin x) + C (103) x cos xdx = cos x + x sin x + C (88) 2 1 x 1 x x cos axdx = cos ax + sin ax + C (89) xex cos xdx = e (x cos x − sin x + x sin x) + C (104) a2 a 2 Integrals of Hyperbolic Functions 2 2 x cos xdx = 2x cos x + x − 2 sin x + C (90) 1 cosh axdx = sinh ax + C (105) a 2 2 2x cos ax a x − 2 x2 cos axdx = + sin ax + C (91) a2 a3 eax cosh bxdx =  ax  e [a cosh bx − b sinh bx] + C a=b 1  2 a − b2 xn cosxdx = − (i)n+1 [Γ(n + 1, −ix) 2ax (106) 2 e x  + +C a=b +(−1)n Γ(n + 1, ix)] + C (92) 4a 2 1 sinh axdx = cosh ax + C (107) a 1 xn cosaxdx = (ia)1−n [(−1)n Γ(n + 1, −iax) 2 eax sinh bxdx = −Γ(n + 1, ixa)] + C (93)  ax  e  2 [−b cosh bx + a sinh bx] + C a=b a − b2 (108) 2ax e x x sin xdx = −x cos x + sin x + C (94)  − +C a=b 4a 2 x cos ax sin ax x sin axdx = − + +C (95) eax tanh bxdx = a a2  (a+2b)x e a a 2bx  (a + 2b) 2 F1 1 + 2b , 1, 2 + 2b , −e  x2 sin xdx = 2 − x2 cos x + 2x sin x + C  (96)   1 a − eax 2 F1 , 1, 1E, −e2bx + C a = b (109)  a −1 ax 2b  eax − 2 tan [e ]   2 − a2 x2 +C a=b  2x sin ax  x2 sin axdx = 3 cos ax + +C (97) a a a3 1 tanh bxdx = ln cosh ax + C (110) a 1 xn sin xdx = − (i)n [Γ(n + 1, −ix) 1 2 cos ax cosh bxdx = [a sin ax cosh bx −(−1)n Γ(n + 1, −ix)] + C (98) a2 + b2 +b cos ax sinh bx] + C (111) 4
  5. 5. 1 1 cos ax sinh bxdx = [b cos ax cosh bx+ sinh ax cosh axdx = [−2ax + sinh 2ax] + C (115) a2+ b2 4a a sin ax sinh bx] + C (112) 1 1 sinh ax cosh bxdx = [b cosh bx sinh axsin ax cosh bxdx = [−a cos ax cosh bx+ b2 − a2 a2 + b2 −a cosh ax sinh bx] + C (116) b sin ax sinh bx] + C (113) 1 sin ax sinh bxdx = [b cosh bx sin ax− a2 + b2 a cos ax sinh bx] + C (114) 5

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