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Table of Basic Integrals

Basic Forms

x n d x = 1 n + 1 x n + 1 , n 1 (1)
1 x d x = ln | x | (2)
u d v = u v v d u (3)
1 a x + b d x = 1 a ln | a x + b | (4)

Integrals of Rational Functions

1 ( x + a ) 2 d x = 1 x + a (5)
( x + a ) n d x = ( x + a ) n + 1 n + 1 , n 1 (6)
x ( x + a ) n d x = ( x + a ) n + 1 ( ( n + 1 ) x a ) ( n + 1 ) ( n + 2 ) (7)
1 1 + x 2 d x = tan 1 x (8)
1 a 2 + x 2 d x = 1 a tan 1 x a (9)
x a 2 + x 2 d x = 1 2 ln | a 2 + x 2 | (10)
x 2 a 2 + x 2 d x = x a tan 1 x a (11)
x 3 a 2 + x 2 d x = 1 2 x 2 1 2 a 2 ln | a 2 + x 2 | (12)
1 a x 2 + b x + c d x = 2 4 a c b 2 tan 1 2 a x + b 4 a c b 2 (13)
1 ( x + a ) ( x + b ) d x = 1 b a ln a + x b + x ,   a b (14)
x ( x + a ) 2 d x = a a + x + ln | a + x | (15)
x a x 2 + b x + c d x = 1 2 a ln | a x 2 + b x + c | b a 4 a c b 2 tan 1 2 a x + b 4 a c b 2 (16)

Integrals with Roots

x a d x = 2 3 ( x a ) 3 2 (17)
1 x ± a d x = 2 x ± a (18)
1 a x d x = 2 a x (19)
x x a d x = 2 a 3 x a 3 2 + 2 5 x a 5 2 ,  or 2 3 x ( x a ) 3 2 4 1 5 ( x a ) 5 2 ,  or 2 1 5 ( 2 a + 3 x ) ( x a ) 3 2 (20)
a x + b d x = 2 b 3 a + 2 x 3 a x + b (21)
( a x + b ) 3 2 d x = 2 5 a ( a x + b ) 5 2 (22)
x x ± a d x = 2 3 ( x 2 a ) x ± a (23)
x a x d x = x ( a x ) a tan 1 x ( a x ) x a (24)
x a + x d x = x ( a + x ) a ln x + x + a (25)
x a x + b d x = 2 1 5 a 2 ( 2 b 2 + a b x + 3 a 2 x 2 ) a x + b (26)
x ( a x + b ) d x = 1 4 a 3 2 ( 2 a x + b ) a x ( a x + b ) b 2 ln a x + a ( a x + b ) (27)
x 3 ( a x + b ) d x = b 1 2 a b 2 8 a 2 x + x 3 x 3 ( a x + b ) + b 3 8 a 5 2 ln a x + a ( a x + b ) (28)
x 2 ± a 2 d x = 1 2 x x 2 ± a 2 ± 1 2 a 2 ln x + x 2 ± a 2 (29)
a 2 x 2 d x = 1 2 x a 2 x 2 + 1 2 a 2 tan 1 x a 2 x 2 (30)
x x 2 ± a 2 d x = 1 3 x 2 ± a 2 3 2 (31)
1 x 2 ± a 2 d x = ln x + x 2 ± a 2 (32)
1 a 2 x 2 d x = sin 1 x a (33)
x x 2 ± a 2 d x = x 2 ± a 2 (34)
x a 2 x 2 d x = a 2 x 2 (35)
x 2 x 2 ± a 2 d x = 1 2 x x 2 ± a 2 1 2 a 2 ln x + x 2 ± a 2 (36)
a x 2 + b x + c d x = b + 2 a x 4 a a x 2 + b x + c + 4 a c b 2 8 a 3 2 ln 2 a x + b + 2 a ( a x 2 + b x + c ) (37)
x a x 2 + b x + c d x = 1 4 8 a 5 2 2 a a x 2 + b x + c 3 b 2 + 2 a b x + 8 a ( c + a x 2 ) + 3 ( b 3 4 a b c ) ln b + 2 a x + 2 a a x 2 + b x + c (38)
1 a x 2 + b x + c d x = 1 a ln 2 a x + b + 2 a ( a x 2 + b x + c ) (39)
x a x 2 + b x + c d x = 1 a a x 2 + b x + c b 2 a 3 2 ln 2 a x + b + 2 a ( a x 2 + b x + c ) (40)
d x ( a 2 + x 2 ) 3 2 = x a 2 a 2 + x 2 (41)

Integrals with Logarithms

ln a x d x = x ln a x x (42)
x ln x d x = 1 2 x 2 ln x x 2 4 (43)
x 2 ln x d x = 1 3 x 3 ln x x 3 9 (44)
x n ln x d x = x n + 1 ln x n + 1 1 ( n + 1 ) 2 , n 1 (45)
ln a x x d x = 1 2 ln a x 2 (46)
ln x x 2 d x = 1 x ln x x (47)
ln ( a x + b ) d x = x + b a ln ( a x + b ) x , a 0 (48)
ln ( x 2 + a 2 ) d x = x ln ( x 2 + a 2 ) + 2 a tan 1 x a 2 x (49)
ln ( x 2 a 2 ) d x = x ln ( x 2 a 2 ) + a ln x + a x a 2 x (50)
ln a x 2 + b x + c d x = 1 a 4 a c b 2 tan 1 2 a x + b 4 a c b 2 2 x + b 2 a + x ln a x 2 + b x + c (51)
x ln ( a x + b ) d x = b x 2 a 1 4 x 2 + 1 2 x 2 b 2 a 2 ln ( a x + b ) (52)
x ln a 2 b 2 x 2 d x = 1 2 x 2 + 1 2 x 2 a 2 b 2 ln a 2 b 2 x 2 (53)
( ln x ) 2 d x = 2 x 2 x ln x + x ( ln x ) 2 (54)
( ln x ) 3 d x = 6 x + x ( ln x ) 3 3 x ( ln x ) 2 + 6 x ln x (55)
x ( ln x ) 2 d x = x 2 4 + 1 2 x 2 ( ln x ) 2 1 2 x 2 ln x (56)
x 2 ( ln x ) 2 d x = 2 x 3 2 7 + 1 3 x 3 ( ln x ) 2 2 9 x 3 ln x (57)

Integrals with Exponentials

e a x d x = 1 a e a x (58)
x e a x d x = 1 a x e a x + i π 2 a 3 2 erf i a x ,  where erf ( x ) = 2 π 0 x e t 2 d t (59)
x e x d x = ( x 1 ) e x (60)
x e a x d x = x a 1 a 2 e a x (61)
x 2 e x d x = x 2 2 x + 2 e x (62)
x 2 e a x d x = x 2 a 2 x a 2 + 2 a 3 e a x (63)
x 3 e x d x = x 3 3 x 2 + 6 x 6 e x (64)
x n e a x d x = x n e a x a n a x n 1 e a x d x (65)
x n e a x d x = ( 1 ) n a n + 1 Γ [ 1 + n , a x ] ,  where  Γ ( a , x ) = x t a 1 e t d t (66)
e a x 2 d x = i π 2 a erf i x a (67)
e a x 2 d x = π 2 a erf x a (68)
x e a x 2 d x = 1 2 a e a x 2 (69)
x 2 e a x 2 d x = 1 4 π a 3 erf ( x a ) x 2 a e a x 2 (70)

Integrals with Trigonometric Functions

sin a x d x = 1 a cos a x (71)
sin 2 a x d x = x 2 sin 2 a x 4 a (72)
sin 3 a x d x = 3 cos a x 4 a + cos 3 a x 1 2 a (73)
sin n a x d x = 1 a cos a x 2 F 1 1 2 , 1 n 2 , 3 2 , cos 2 a x (74)
cos a x d x = 1 a sin a x (75)
cos 2 a x d x = x 2 + sin 2 a x 4 a (76)
cos 3 a x d x = 3 sin a x 4 a + sin 3 a x 1 2 a (77)
cos p a x d x = 1 a ( 1 + p ) cos 1 + p a x × 2 F 1 1 + p 2 , 1 2 , 3 + p 2 , cos 2 a x (78)
cos x sin x d x = 1 2 sin 2 x + c 1 = 1 2 cos 2 x + c 2 = 1 4 cos 2 x + c 3 (79)
cos a x sin b x d x = cos [ ( a b ) x ] 2 ( a b ) cos [ ( a + b ) x ] 2 ( a + b ) , a b (80)
sin 2 a x cos b x d x = sin [ ( 2 a b ) x ] 4 ( 2 a b ) + sin b x 2 b sin [ ( 2 a + b ) x ] 4 ( 2 a + b ) (81)
sin 2 x cos x d x = 1 3 sin 3 x (82)
cos 2 a x sin b x d x = cos [ ( 2 a b ) x ] 4 ( 2 a b ) cos b x 2 b cos [ ( 2 a + b ) x ] 4 ( 2 a + b ) (83)
cos 2 a x sin a x d x = 1 3 a cos 3 a x (84)
sin 2 a x cos 2 b x d x = x 4 sin 2 a x 8 a sin [ 2 ( a b ) x ] 1 6 ( a b ) + sin 2 b x 8 b sin [ 2 ( a + b ) x ] 1 6 ( a + b ) (85)
sin 2 a x cos 2 a x d x = x 8 sin 4 a x 3 2 a (86)
tan a x d x = 1 a ln cos a x (87)
tan 2 a x d x = x + 1 a tan a x (88)
tan n a x d x = tan n + 1 a x a ( 1 + n ) × 2 F 1 n + 1 2 , 1 , n + 3 2 , tan 2 a x (89)
tan 3 a x d x = 1 a ln cos a x + 1 2 a sec 2 a x (90)
sec x d x = ln | sec x + tan x | = 2 tanh 1 tan x 2 (91)
sec 2 a x d x = 1 a tan a x (92)
sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | (93)
sec x tan x d x = sec x (94)
sec 2 x tan x d x = 1 2 sec 2 x (95)
sec n x tan x d x = 1 n sec n x , n 0 (96)
csc x d x = ln tan x 2 = ln | csc x cot x | + C (97)
csc 2 a x d x = 1 a cot a x (98)
csc 3 x d x = 1 2 cot x csc x + 1 2 ln | csc x cot x | (99)
csc n x cot x d x = 1 n csc n x , n 0 (100)
sec x csc x d x = ln | tan x | (101)

Products of Trigonometric Functions and Monomials

x cos x d x = cos x + x sin x (102)
x cos a x d x = 1 a 2 cos a x + x a sin a x (103)
x 2 cos x d x = 2 x cos x + x 2 2 sin x (104)
x 2 cos a x d x = 2 x cos a x a 2 + a 2 x 2 2 a 3 sin a x (105)
x n cos x d x = 1 2 ( i ) n + 1 Γ ( n + 1 , i x ) + ( 1 ) n Γ ( n + 1 , i x ) (106)
x n cos a x d x = 1 2 ( i a ) 1 n ( 1 ) n Γ ( n + 1 , i a x ) Γ ( n + 1 , i x a ) (107)
x sin x d x = x cos x + sin x (108)
x sin a x d x = x cos a x a + sin a x a 2 (109)
x 2 sin x d x = 2 x 2 cos x + 2 x sin x (110)
x 2 sin a x d x = 2 a 2 x 2 a 3 cos a x + 2 x sin a x a 2 (111)
x n sin x d x = 1 2 ( i ) n Γ ( n + 1 , i x ) ( 1 ) n Γ ( n + 1 , i x ) (112)
x cos 2 x d x = x 2 4 + 1 8 cos 2 x + 1 4 x sin 2 x (113)
x sin 2 x d x = x 2 4 1 8 cos 2 x 1 4 x sin 2 x (114)
x tan 2 x d x = x 2 2 + ln cos x + x tan x (115)
x sec 2 x d x = ln cos x + x tan x (116)

Products of Trigonometric Functions and Exponentials

e x sin x d x = 1 2 e x ( sin x cos x ) (117)
e b x sin a x d x = 1 a 2 + b 2 e b x ( b sin a x a cos a x ) (118)
e x cos x d x = 1 2 e x ( sin x + cos x ) (119)
e b x cos a x d x = 1 a 2 + b 2 e b x ( a sin a x + b cos a x ) (120)
x e x sin x d x = 1 2 e x ( cos x x cos x + x sin x ) (121)
x e x cos x d x = 1 2 e x ( x cos x sin x + x sin x ) (122)

Integrals of Hyperbolic Functions

cosh a x d x = 1 a sinh a x (123)
e a x cosh b x d x = e a x a 2 b 2 [ a cosh b x b sinh b x ] a b e 2 a x 4 a + x 2 a = b (124)
sinh a x d x = 1 a cosh a x (125)
e a x sinh b x d x = e a x a 2 b 2 [ b cosh b x + a sinh b x ] a b e 2 a x 4 a x 2 a = b (126)
tanh a x d x = 1 a ln cosh a x (127)
e a x tanh b x d x = e ( a + 2 b ) x ( a + 2 b ) ( 2 F 1 ) 1 + a 2 b , 1 , 2 + a 2 b , e 2 b x e a x a ( 2 F 1 ) 1 , a 2 b , 1 + a 2 b , e 2 b x a b e a x 2 tan 1 [ e a x ] a a = b (128)
cos a x cosh b x d x = 1 a 2 + b 2 a sin a x cosh b x + b cos a x sinh b x (129)
cos a x sinh b x d x = 1 a 2 + b 2 b cos a x cosh b x + a sin a x sinh b x (130)
sin a x cosh b x d x = 1 a 2 + b 2 a cos a x cosh b x + b sin a x sinh b x (131)
sin a x sinh b x d x = 1 a 2 + b 2 b cosh b x sin a x a cos a x sinh b x (132)
sinh a x cosh a x d x = 1 4 a 2 a x + sinh 2 a x (133)
sinh a x cosh b x d x = 1 b 2 a 2 b cosh b x sinh a x a cosh a x sinh b x (134)

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