This document provides formulas for integrals of common functions. It includes integrals of polynomials, rational functions, radicals, logarithms, exponentials, trigonometric functions, and their combinations. There are a total of 94 formulas organized into sections based on function type. The integrals cover basic forms, rational functions, radicals, logarithms, exponentials, and trigonometric functions.

1. Table of Basic Integrals
Basic Forms
(1)
Z
xndx =
1
n + 1
xn+1; n6= 1
(2)
Z
1
x
dx = ln jxj
(3)
Z
udv = uv
Z
vdu
(4)
Z
1
ax + b
dx =
1
a
ln jax + bj
Integrals of Rational Functions
(5)
Z
1
(x + a)2 dx =
1
x + a
(6)
Z
(x + a)ndx =
(x + a)n+1
n + 1
; n6= 1
(7)
Z
x(x + a)ndx =
(x + a)n+1((n + 1)x a)
(n + 1)(n + 2)
(8)
Z
1
1 + x2 dx = tan1 x
(9)
Z
1
a2 + x2 dx =
1
a
tan1 x
a
1

2. (10)
Z
x
a2 + x2 dx =
1
2
ln ja2 + x2j
(11)
Z
x2
x
dx = x a tan1 a2 + x2 a
(12)
Z
x3
a2 + x2 dx =
1
2
x2
1
2
a2 ln ja2 + x2j
(13)
Z
1
ax2 + bx + c
dx =
2
p
4ac b2
tan1 2ax + b
p
4ac b2
(14)
Z
1
(x + a)(x + b)
dx =
1
b a
ln
a + x
b + x
; a6= b
(15)
Z
x
(x + a)2 dx =
a
a + x
+ ln ja + xj
(16)
Z
x
ax2 + bx + c
dx =
1
2a
ln jax2+bx+cj
b
p
4ac b2
a
tan1 2ax + b
p
4ac b2
Integrals with Roots
(17)
Z
p
x a dx =
2
3
(x a)3=2
(18)
Z
1
p
x a
p
x a
dx = 2
(19)
Z
1
p
a x
p
a x
dx = 2
2

3. (20)
Z
x
p
x a dx =
8
:
2a
3 (x a)3=2 + 2
5 (x a)5=2 ; or
2
3x(x a)3=2 4
15 (x a)5=2; or
2
15(2a + 3x)(x a)3=2
(21)
Z p
ax + b dx =
2b
3a
+
2x
3
p
ax + b
(22)
Z
(ax + b)3=2 dx =
2
5a
(ax + b)5=2
(23)
Z
x
p
x a
dx =
2
3
p
x a
(x 2a)
(24)
Z r
x
a x
p
x(a x) a tan1
dx =
p
x(a x)
x a
(25)
Z r
x
a + x
dx =
p
x(a + x) a ln
p
x +
p
x + a
(26)
Z
x
p
ax + b dx =
2
15a2 (2b2 + abx + 3a2x2)
p
ax + b
(Z27)p
x(ax + b) dx =
1
4a3=2
h
(2ax + b)
p
ax(ax + b) b2 ln

6. p
x +
a

9. p
a(ax + b)
i
(Z28)p
x3(ax + b) dx =
b
12a
b2
8a2x
+
x
3
p
x3(ax + b)+
b3
8a5=2 ln

12. p
x +
a
p
a(ax + b)

15. (29)
Z p
x2 a2 dx =
1
2
x
p
x2 a2
1
2
a2 ln

18. x +
p
x2 a2

21. 3

22. (30)
Z p
a2 x2 dx =
1
2
x
p
a2 x2 +
1
2
a2 tan1 x
p
a2 x2
(31)
Z
x
p
x2 a2 dx =
1
3
x2 a23=2
(32)
Z
1
p
x2 a2
dx = ln

25. x +
p
x2 a2

28. (33)
Z
1
p
a2 x2
dx = sin1 x
a
(34)
Z
x
p
x2 a2
dx =
p
x2 a2
(35)
Z
x
p
a2 x2
p
a2 x2
dx =
(36)
Z
x2
p
x2 a2
dx =
1
2
x
p
x2 a2
1
2
a2 ln

31. x +
p
x2 a2

34. (Z37)p
ax2 + bx + c dx =
b + 2ax
4a
p
ax2 + bx + c+
4ac b2
8a3=2 ln

37. 2ax + b + 2

40. p
a(ax2 + bx+c)
Z
x
p
ax2 + bx + c dx =
1
48a5=2
2
p
ax2 + bx + c
p
a
3b2 + 2abx + 8a(c + ax2)
+3(b3 4abc) ln

43. b + 2ax + 2
p
ax2 + bx + c
p
a

46. (38)
4

47. (39)
Z
1
p
ax2 + bx + c
dx =
1
p
a
ln

50. 2ax + b + 2
p
a(ax2 + bx + c)

53. (Z40)
x
p
ax2 + bx + c
dx =
1
a
p
ax2 + bx + c
b
2a3=2 ln

56. 2ax + b + 2
p
a(ax2 + bx + c)

59. (41)
Z
dx
(a2 + x2)3=2 =
x
a2
p
a2 + x2
Integrals with Logarithms
(42)
Z
ln ax dx = x ln ax x
(43)
Z
x ln x dx =
1
2
x2 ln x
x2
4
(44)
Z
x2 ln x dx =
1
3
x3 ln x
x3
9
(45)
Z
xn ln x dx = xn+1
ln x
n + 1
1
(n + 1)2
; n6= 1
(46)
Z
ln ax
x
dx =
1
2
(ln ax)2
(47)
Z
ln x
x2 dx =
1
x
ln x
x
5

60. (48)
Z
ln(ax + b) dx =
x +
b
a
ln(ax + b) x; a6= 0
(49)
Z
ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan1 x
a
2x
(50)
Z
ln(x2 a2) dx = x ln(x2 a2) + a ln
x + a
x a
2x
(Z51)
ln
ax2 + bx + c
dx =
1
a
p
4ac b2 tan1 2ax + b
p
4ac b2
2x+
b
2a
+ x
ln
ax2 + bx + c
(52)
Z
x ln(ax + b) dx =
bx
2a
1
4
x2 +
1
2
x2
b2
a2
ln(ax + b)
(53)
Z
x ln
a2 b2x2
dx =
1
2
x2 +
1
2
x2
a2
b2
ln
a2 b2x2
(54)
Z
(ln x)2 dx = 2x 2x ln x + x(ln x)2
(55)
Z
(ln x)3 dx = 6x + x(ln x)3 3x(ln x)2 + 6x ln x
(56)
Z
x(ln x)2 dx =
x2
4
+
1
2
x2(ln x)2
1
2
x2 ln x
(57)
Z
x2(ln x)2 dx =
2x3
27
+
1
3
x3(ln x)2
2
9
x3 ln x
6

61. Integrals with Exponentials
(58)
Z
eax dx =
1
a
eax
(5Z9)
p
xeax dx =
1
a
p
xeax +
p
i
erf
2a3=2
i
p
ax
; where erf(x) =
2
p
Z x
0
et2
dt
(60)
Z
xex dx = (x 1)ex
(61)
Z
xeax dx =
x
a
1
a2
eax
(62)
Z
x2ex dx =
x2 2x + 2
ex
(63)
Z
x2eax dx =
x2
a
2x
a2 +
2
a3
eax
(64)
Z
x3ex dx =
x3 3x2 + 6x 6
ex
(65)
Z
xneax dx =
xneax
a
n
a
Z
xn1eaxdx
(66)
Z
xneax dx =
(1)n
an+1 [1 + n;ax]; where (a; x) =
Z 1
x
ta1et dt
(67)
Z
eax2
dx =
p
i
2
p
a
erf
ix
p
a
7

62. (68)
Z
eax2
dx =
p
2
p
a
erf
x
p
a
(69)
Z
xeax2
dx =
1
2a
eax2
(70)
Z
x2eax2
dx =
1
4
r
a3 erf(x
p
a)
x
2a
eax2
Integrals with Trigonometric Functions
(71)
Z
sin ax dx =
1
a
cos ax
(72)
Z
sin2 ax dx =
x
2
sin 2ax
4a
(73)
Z
sin3 ax dx =
3 cos ax
4a
+
cos 3ax
12a
(74)
Z
sinn ax dx =
1
a
cos ax 2F1
1
2
;
1 n
2
;
3
2
; cos2 ax
(75)
Z
cos ax dx =
1
a
sin ax
(76)
Z
cos2 ax dx =
x
2
+
sin 2ax
4a
(77)
Z
cos3 axdx =
3 sin ax
4a
+
sin 3ax
12a
8

63. (78)
Z
cosp axdx =
1
a(1 + p)
cos1+p ax 2F1
1 + p
2
;
1
2
;
3 + p
2
; cos2 ax
(79)
Z
cos x sin x dx =
1
2
sin2 x + c1 =
1
2
cos2 x + c2 =
1
4
cos 2x + c3
(80)
Z
cos ax sin bx dx =
cos[(a b)x]
2(a b)
cos[(a + b)x]
2(a + b)
; a6= b
(81)
Z
sin2 ax cos bx dx =
sin[(2a b)x]
4(2a b)
+
sin bx
2b
sin[(2a + b)x]
4(2a + b)
(82)
Z
sin2 x cos x dx =
1
3
sin3 x
(83)
Z
cos2 ax sin bx dx =
cos[(2a b)x]
4(2a b)
cos bx
2b
cos[(2a + b)x]
4(2a + b)
(84)
Z
cos2 ax sin ax dx =
1
3a
cos3 ax
(Z85)
sin2 ax cos2 bxdx =
x
4
sin 2ax
8a
sin[2(a b)x]
16(a b)
+
sin 2bx
8b
sin[2(a + b)x]
16(a + b)
(86)
Z
sin2 ax cos2 ax dx =
x
8
sin 4ax
32a
(87)
Z
tan ax dx =
1
a
ln cos ax
9

64. (88)
Z
tan2 ax dx = x +
1
a
tan ax
(89)
Z
tann ax dx =
tann+1 ax
a(1 + n)
2F1
n + 1
2
; 1;
n + 3
2
;tan2 ax
(90)
Z
tan3 axdx =
1
a
ln cos ax +
1
2a
sec2 ax
(91)
Z
sec x dx = ln j sec x + tan xj = 2 tanh1
tan
x
2
(92)
Z
sec2 ax dx =
1
a
tan ax
(93)
Z
sec3 x dx =
1
2
sec x tan x +
1
2
ln j sec x + tan xj
(94)
Z
sec x tan x dx = sec x
(95)
Z
sec2 x tan x dx =
1
2
sec2 x
(96)
Z
secn x tan x dx =
1
n
secn x; n6= 0
(97)
Z
csc x dx = ln

67. tan
x
2

70. = ln j csc x cot xj + C
10

71. (98)
Z
csc2 ax dx =
1
a
cot ax
(99)
Z
csc3 x dx =
1
2
cot x csc x +
1
2
ln j csc x cot xj
(100)
Z
cscn x cot x dx =
1
n
cscn x; n6= 0
(101)
Z
sec x csc x dx = ln j tan xj
Products of Trigonometric Functions and Mono-
mials
(102)
Z
x cos x dx = cos x + x sin x
(103)
Z
x cos ax dx =
1
a2 cos ax +
x
a
sin ax
(104)
Z
x2 cos x dx = 2x cos x +
x2 2
sin x
(105)
Z
x2 cos ax dx =
2x cos ax
a2 +
a2x2 2
a3 sin ax
(106)
Z
xn cos xdx =
1
2
(i)n+1 [(n + 1;ix) + (1)n(n + 1; ix)]
11

72. (107)
Z
xn cos ax dx =
1
2
(ia)1n [(1)n(n + 1;iax) (n + 1; ixa)]
(108)
Z
x sin x dx = x cos x + sin x
(109)
Z
x sin ax dx =
x cos ax
a
+
sin ax
a2
(110)
Z
x2 sin x dx =
2 x2
cos x + 2x sin x
(111)
Z
x2 sin ax dx =
2 a2x2
a3 cos ax +
2x sin ax
a2
(112)
Z
xn sin x dx =
1
2
(i)n [(n + 1;ix) (1)n(n + 1;ix)]
(113)
Z
x cos2 x dx =
x2
4
+
1
8
cos 2x +
1
4
x sin 2x
(114)
Z
x sin2 x dx =
x2
4
1
8
cos 2x
1
4
x sin 2x
(115)
Z
x tan2 x dx =
x2
2
+ ln cos x + x tan x
(116)
Z
x sec2 x dx = ln cos x + x tan x
12

73. Products of Trigonometric Functions and Ex-
ponentials
(117)
Z
ex sin x dx =
1
2
ex(sin x cos x)
(118)
Z
ebx sin ax dx =
1
a2 + b2 ebx(b sin ax a cos ax)
(119)
Z
ex cos x dx =
1
2
ex(sin x + cos x)
(120)
Z
ebx cos ax dx =
1
a2 + b2 ebx(a sin ax + b cos ax)
(121)
Z
xex sin x dx =
1
2
ex(cos x x cos x + x sin x)
(122)
Z
xex cos x dx =
1
2
ex(x cos x sin x + x sin x)
Integrals of Hyperbolic Functions
(123)
Z
cosh ax dx =
1
a
sinh ax
(124)
Z
eax cosh bx dx =
8
:
eax
[a cosh bx b sinh bx] a6= b
a2 b2 e2ax
x
+
a = b
4a
2
(125)
Z
sinh ax dx =
1
a
cosh ax
13

74. (126)
Z
eax sinh bx dx =
8
:
eax
[b cosh bx + a sinh bx] a6= b
a2 b2 e2ax
x
a = b
4a
2
(127)
Z
tanh axdx =
1
a
ln cosh ax
(128)
Z
eax tanh bx dx =
8
:
e(a+2b)x
(a + 2b) 2F1
h
1 +
a
2b
; 1; 2 +
a
2b
;e2bx
i
1
a
eax
2F1
h
1;
a
2b
; 1 +
a
2b
;e2bx
i
a6= b
eax 2 tan1[eax]
a
a = b
(129)
Z
cos ax cosh bx dx =
1
a2 + b2 [a sin ax cosh bx + b cos ax sinh bx]
(130)
Z
cos ax sinh bx dx =
1
a2 + b2 [b cos ax cosh bx + a sin ax sinh bx]
(131)
Z
sin ax cosh bx dx =
1
a2 + b2 [a cos ax cosh bx + b sin ax sinh bx]
(132)
Z
sin ax sinh bx dx =
1
a2 + b2 [b cosh bx sin ax a cos ax sinh bx]
(133)
Z
sinh ax cosh axdx =
1
4a
[2ax + sinh 2ax]
(134)
Z
sinh ax cosh bx dx =
1
b2 a2 [b cosh bx sinh ax a cosh ax sinh bx]
c 2014. From http://integral-table.com, last revised June 14, 2014. This mate-
rial is provided as is without warranty or representation about the accuracy, correctness or
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