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Table of Integrals∗ Basic Forms Integrals with Logarithms √ x ax + bdx = x dx = xn+1 n+1 (1) dx = ln |x| x (2) n udv = uv − vdu x(ax + b)dx = 3/2 (2ax + b) ax(ax + b) 4a √ (27) −b2 ln a x + a(ax + b) (3) 1 dx = ln |ax + b| ax + b a 1 dx = − (x + a)2 x+a x(x + a)n dx = b b2 x − + x3 (ax + b) 12a 8a x √ b3 (28) + 5/2 ln a x + a(ax + b) 8a x3 (ax + b)dx = (5) (x + a)n+1 , n = −1 n+1 (x + a)n+1 ((n + 1)x − a) (n + 1)(n + 2) (6) 1 x dx = tan−1 a2 + x2 a a (9) a ln x + x2 ± a2 (29) a2 − x2 dx = x (10) x2 x dx = x − a tan−1 a2 + x2 a x2 ± a2 ± (7) (8) x dx = ln |a2 + x2 | a2 + x2 x x2 ± a2 dx = dx = tan−1 x + x2 x a2 − x2 + x a tan−1 √ a2 − x2 (30) x2 ± a2 dx = x ± a2 √ dx = ln x + x2 ± a2 (11) 3/2 x 1 dx = x2 − a2 ln |a2 + x2 | a2 + x2 2 (12) 2ax + b dx = √ tan−1 √ ax2 + bx + c 4ac − b 4ac − b2 (13) 1 a+x dx = ln , a=b (x + a)(x + b) b−a b+x (14) x a dx = + ln |a + x| (x + a)2 a+x x dx = ln |ax2 + bx + c| ax2 + bx + c 2a b 2ax + b − √ tan−1 √ a 4ac − b2 4ac − b2 x2 ± a2 (32) x √ dx = sin−1 a a2 − x2 x √ dx = x2 ± a2 √ x − adx = (x − a)3/2 a−x ax + bdx = √ dx = x ± a x2 √ dx = x 2 x ±a (19) √ (21) √ x dx = (x ∓ 2a) x ± a x±a (23) x dx = a+x x(a + x) − a ln √ x+ ln(ax + b) − x, a = (44) x(a − x) x−a √ x+a a2 − x2 x2 ± a2 ∓ a2 ln x + (24) (25) a (35) x ln a2 − b2 x2 dx = − x2 + a2 x − 2 b eax dx = √ ax e a (50) √ √ √ ax i π xe + 3/2 erf i ax , a 2a x 2 where erf(x) = √ e−t dt π xex dx = (x − 1)ex (51) x − a a eax x2 2x − + a a a (53) (54) (55) x3 ex dx = x3 − 3x2 + 6x − ex (56) (38) xn eax n − a a + bx + c) where Γ(a, x) = (57) (58) ta−1 e−t dt x √ √ i π eax dx = − √ erf ix a a √ √ π e−ax dx = √ erf x a a 2 x2 e−ax dx = (59) (60) −ax2 e 2a (61) √ π x −ax2 erf(x a) − e a3 2a (62) xe−ax dx = − (40) (41) xn−1 eax dx (−1)n Γ[1 + n, −ax], an+1 ∞ a(ax2 (52) eax xn eax dx = dx x = √ (a2 + x2 )3/2 a2 a2 + x2 (49) (37) + bx + c x √ dx = ax2 + bx + c a ax2 + bx + c b − 3/2 ln 2ax + b + a(ax2 + bx + c) 2a ln a2 − b2 x2 xeax dx = xn eax dx = 1 √ dx = √ ln 2ax + b + 2 a ax + bx + c (48) x2 ex dx = x2 − 2x + ex ax2 + bx + c ax2 ln(ax + b) Integrals with Exponentials x2 eax dx = × −3b2 + 2abx + 8a(c + ax2 ) √ +3(b3 − 4abc) ln b + 2ax + a (47) x2 ± a2 b + 2ax ax2 + bx + cdx = ax2 + bx + c 4a 4ac − b2 + ln 2ax + b + a(ax2 + bx+ c) 8a3/2 √ a 5/2 48a x+a − 2x (46) x−a 2ax + b 4ac − b2 tan−1 √ 4ac − b2 bx − x2 2a b2 + x2 − 2 a xeax dx = ax2 + bx + c = x − 2x (45) a x ln(ax + b)dx = (39) (22) x(a − x) − a tan−1 b a b + x ln ax2 + bx + c 2a − 2x + (34) (20) (ax + b)5/2 5a x dx = − a−x ∗ √ ax + b ln ax2 + bx + c dx = (16) (18) √ dx = −2 a − x (ax + b)3/2 dx = x+ ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln (36) x 2b 2x + 3a (43) ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1 (15) (17) √ 2 x x − adx = a(x − a)3/2 + (x − a)5/2 √ ln ax dx = (ln ax)2 x ln(ax + b)dx = (33) x2 ± a2 x √ dx = − a2 − x2 Integrals with Roots √ (42) (31) √ x±a ln axdx = x ln ax − x (4) Integrals of Rational Functions (x + a)n dx = √ (−2b2 + abx + 3a2 x2 ) ax + b (26) 15a2 2014 From http://integral-table.com, last revised June 14, 2014 This material is provided as is without warranty or representation about the accuracy, correctness or suitability of the material for any purpose, and is licensed under the Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA Integrals with Trigonometric Functions sec3 x dx = sin axdx = − cos ax a sin2 axdx = 1 sec x tan x + ln | sec x + tan x| 2 sec x tan xdx = sec x x sin 2ax − 4a sin axdx = sec x tan xdx = secn x, n = n 1−n , , , cos2 ax 2 (66) sin ax a cos ax sin bxdx = (68) 1 csc3 xdx = − cot x csc x + ln | csc x − cot x| 2 csc x cot xdx = − cscn x, n = n n sec x csc xdx = ln | tan x| (69) (70) ebx (a sin ax + b cos ax) a2 + b2 (107) xex sin xdx = x e (cos x − x cos x + x sin x) (108) xex cos xdx = x e (x cos x − sin x + x sin x) (109) (86) (87) (88) (89) x cos xdx = cos x + x sin x x cos ax + sin ax a2 a x cos axdx = sin[(2a − b)x] sin ax cos bxdx = − 4(2a − b) sin[(2a + b)x] sin bx − + 2b 4(2a + b) (91) (92) (93) (94) (95) (72) 2x cos ax a2 x2 − x cos axdx = + sin ax a a3 sin3 x (73) cos[(2a − b)x] cos bx cos ax sin bxdx = − 4(2a − b) 2b cos[(2a + b)x] − 4(2a + b) xn cosxdx = − (i)n+1 [Γ(n + 1, −ix) +(−1)n Γ(n + 1, ix)] cos3 ax 3a eax cosh bxdx =  ax e   [a cosh bx − b sinh bx] a − b2 2ax x  e + 4a sinh axdx = x2 cos xdx = 2x cos x + x2 − sin x (96) (97) eax bxdx =  (a+2b)x  e a a   , 1, + , −e2bx F1 +   2b 2b  (a + 2b) a , 1, 1E, −e2bx − eax F1  a 2b   eax − tan−1 [eax ]    a ax dx = sin[2(a − b)x] x sin 2ax − − 8a 16(a − b) sin[2(a + b)x] sin 2bx + − (76) 8b 16(a + b) (ia)1−n [(−1)n Γ(n + 1, −iax) −Γ(n + 1, ixa)] (98) sin 4ax x − 32a tan2 axdx = −x + tan ax a (79) (80) 1 ln cos ax + sec2 ax a 2a sec xdx = ln | sec x + tan x| = tanh−1 tan tan ax a a=b (113) a=b a = b (114) a=b ln cosh ax a (115) [a sin ax cosh bx a2 + b2 +b cos ax sinh bx] (116) cos ax cosh bxdx = sin ax x cos ax + a a2 (100) x2 sin xdx = − x2 cos x + 2x sin x (101) x sin axdx = − [b cos ax cosh bx+ a2 + b2 a sin ax sinh bx] (117) cos ax sinh bxdx = (81) x2 sin axdx = 2 [−a cos ax cosh bx+ a2 + b2 b sin ax sinh bx] (118) 2−a x 2x sin ax cos ax + a3 a2 (102) xn sin xdx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)] (103) [b cosh bx sin ax− a2 + b2 a cos ax sinh bx] (119) sin ax sinh bxdx = Products of Trigonometric Functions and Exponentials sinh ax cosh axdx = x (82) ex sin xdx = x e (sin x − cos x) (83) ebx sin axdx = ebx (b sin ax − a cos ax) a2 + b2 [−2ax + sinh 2ax] 4a (120) (104) [b cosh bx sinh ax b2 − a2 −a cosh ax sinh bx] (121) sinh ax cosh bxdx = sec2 axdx = (112) sin ax cosh bxdx = tann+1 ax × a(1 + n) n+3 n+1 , 1, , − tan2 ax 2 tan3 axdx = (111) a=b (99) (78) tann axdx = F1 x sin xdx = −x cos x + sin x (77) tan axdx = − ln cos ax a a=b xn cosaxdx = sin2 ax cos2 bxdx = sin2 ax cos2 axdx = (110) cosh ax a eax sinh bxdx =  ax e   [−b cosh bx + a sinh bx] a − b2 2ax x  e − 4a (74) (75) sinh ax a (90) Products of Trigonometric Functions and Monomials cos[(a − b)x] cos[(a + b)x] − ,a = b 2(a − b) 2(a + b) (71) cos2 ax sin axdx = − ebx cos axdx = cosh axdx = sin ax sin 3ax + 4a 12a sin2 x cos xdx = (106) Integrals of Hyperbolic Functions csc axdx = − cot ax a (67) cos1+p ax× a(1 + p) 1+p 3+p , , , cos2 ax 2 cos3 axdx = x = ln | csc x − cot x| + C csc xdx = ln tan 2 sin 2ax x cos2 axdx = + 4a F1 (85) (65) cos ax cos 3ax + 4a 12a cos axdx = cosp axdx = − sec2 x n sin3 axdx = − x e (sin x + cos x) (64) n F1 ex cos xdx = (63) sec2 x tan xdx = − cos ax a (84) (105)