This document provides formulas for integrals involving common functions including: - Logarithms - Roots - Rational functions - Trigonometric functions - Exponentials - Products of functions It includes the basic forms for integrals with these functions as well as more complex integrals combining multiple functions. The integrals are grouped by function type for easy reference.

1. Table of Integrals*
Integrals with Logarithms
Basic Forms
1
x dx =
xn+1
n+1
1
dx = ln |x|
x
(2)
x
dx =
a+x
(1)
n
udv = uv −
vdu
√
x ax + bdx =
x(a + x) − a ln
√
x+
√
x+a
(25)
ln axdx = x ln ax − x
(3)
1
1
dx = ln |ax + b|
ax + b
a
ln ax
1
dx = (ln ax)2
x
2
√
2
(−2b2 + abx + 3a2 x2 ) ax + b (26)
15a2
(42)
(43)
ln(ax + b)dx =
1
x(ax + b)dx = 3/2 (2ax + b) ax(ax + b)
4a
√
(27)
−b2 ln a x + a(ax + b)
(4)
x+
b
a
ln(ax + b) − x, a = 0
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
Integrals of Rational Functions
1
1
dx = −
(x + a)2
x+a
(x + a)n dx =
b
b2
x
− 2 +
x3 (ax + b)
12a
8a x
3
√
b3
+ 5/2 ln a x + a(ax + b)
(28)
8a
x3 (ax + b)dx =
(5)
(x + a)n+1
, n = −1
n+1
(6)
n+1
x(x + a)n dx =
(x + a)
((n + 1)x − a)
(n + 1)(n + 2)
1
x2 ± a2 dx = x
2
(7)
1
x2 ± a2 ± a2 ln x +
2
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
ln ax2 + bx + c dx =
x2 ± a2
(9)
x
1
dx = ln |a2 + x2 |
a2 + x2
2
(10)
a2 − x2 dx =
x
2
a2
x
x
dx = x − a tan−1
+ x2
a
x3
1
1
dx = x2 − a2 ln |a2 + x2 |
2 + x2
a
2
2
1
2
2ax + b
dx = √
tan−1 √
ax2 + bx + c
4ac − b2
4ac − b2
1
1
a+x
dx =
ln
, a=b
(x + a)(x + b)
b−a
b+x
x
a
dx =
+ ln |a + x|
(x + a)2
a+x
ax2
x
1
dx =
ln |ax2 + bx + c|
+ bx + c
2a
2ax + b
b
− √
tan−1 √
a 4ac − b2
4ac − b2
(11)
(13)
x
√
dx =
x2 ± a2
(15)
√
(ax + b)3/2 dx =
√
(16)
(19)
*
«
x
dx = −
a−x
ax2 + bx + cdx =
(33)
eax dx =
(34)
√
a2 − x2
b + 2ax
4a
4ac − b2
ln 2ax + b + 2
8a3/2
xeax dx =
x
1
− 2
a
a
(37)
(24)
eax
(53)
x2
2x
2
− 2 + 3
a
a
a
(54)
eax
(55)
x3 ex dx = x3 − 3x2 + 6x − 6 ex
ax2
x2 eax dx =
(51)
(52)
x2 ex dx = x2 − 2x + 2 ex
(56)
+ bx + c
xn eax dx =
2
ax2 + bx + c
1
1
√
dx = √ ln 2ax + b + 2
a
ax2 + bx + c
xn eax
n
−
a
a
(38)
a(ax2 + bx + c)
dx
x
= √
+ x2 )3/2
a2 a2 + x2
(57)
(58)
ta−1 e−t dt
x
√
√
2
i π
eax dx = − √ erf ix a
2 a
√
√
2
π
e−ax dx = √ erf x a
2 a
2
2
x2 e−ax dx =
1
4
(59)
(60)
1 −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
where Γ(a, x) =
x
1
√
dx =
ax2 + bx + c
a
ax2 + bx + c
b
− 3/2 ln 2ax + b + 2 a(ax2 + bx + c)
2a
(a2
(50)
xex dx = (x − 1)ex
x2 ± a2
(39)
(23)
x(a − x)
x−a
1 ax
e
a
√
√
1 √ ax
i π
xe + 3/2 erf i ax ,
a
2a
x
2
2
where erf(x) = √
e−t dt
π 0
(35)
a(ax2 + bx+ c)
× −3b + 2abx + 8a(c + ax )
√
+3(b3 − 4abc) ln b + 2ax + 2 a
(49)
xeax dx =
ax2 + bx + c
√
1
ax2 + bx + c =
2 a
48a5/2
ln a2 − b2 x2
Integrals with Exponentials
1 2
a ln x +
2
x2 ± a2
(48)
(32)
(21)
√
2a) x ± a
x(a − x) − a tan
x2 ± a2
∞
(22)
−1
(31)
xn eax dx =
(20)
(47)
ln(ax + b)
1
x ln a2 − b2 x2 dx = − x2 +
2
1
a2
x2 − 2
2
b
(36)
2
(ax + b)5/2
5a
2
x
dx = (x
3
x±a
x2
1
dx = x
2
± a2
x2
2
√
ax + b
2b
2x
+
3a
3
3/2
x2 ± a2
x
√
dx = −
a2 − x2
(14)
x
√
1
dx = −2 a − x
a−x
ax + bdx =
1 2
x2 ± a2 dx =
x ± a2
3
(17)
√
2
2
x x − adx = a(x − a)3/2 + (x − a)5/2
3
5
√
1 2
x
a tan−1 √
2
a2 − x2
(30)
a2 − x2 +
1
x
√
dx = sin−1
a
a2 − x2
(18)
2ax + b
4ac − b2 tan−1 √
4ac − b2
bx
1
− x2
2a
4
1
b2
+
x2 − 2
2
a
(12)
+
√
1
√
dx = 2 x ± a
x±a
√
1
x
2
1
√
dx = ln x +
x2 ± a2
Integrals with Roots
√
2
x − adx = (x − a)3/2
3
x+a
− 2x (46)
x−a
x ln(ax + b)dx =
(8)
1
1
x
dx = tan−1
a2 + x2
a
a
x
− 2x (45)
a
b
+ x ln ax2 + bx + c
2a
− 2x +
(29)
1
dx = tan−1 x
1 + x2
1
a
(44)
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USA.

2. Integrals with Trigonometric Functions
sec3 x dx =
1
sin axdx = − cos ax
a
sin2 axdx =
1
1
sec x tan x + ln | sec x + tan x|
2
2
sec x tan xdx = sec x
x
sin 2ax
−
2
4a
sin axdx =
1
sec x tan xdx = secn x, n = 0
n
1 1−n 3
,
, , cos2 ax
2
2
2
(66)
1
sin ax
a
cos ax sin bxdx =
(68)
1
1
csc3 xdx = − cot x csc x + ln | csc x − cot x|
2
2
1
csc x cot xdx = − cscn x, n = 0
n
n
sec x csc xdx = ln | tan x|
(69)
(70)
(107)
1 x
e (cos x − x cos x + x sin x)
2
(108)
xex cos xdx =
1 x
e (x cos x − sin x + x sin x)
2
(109)
(86)
(87)
(88)
(89)
x cos xdx = cos x + x sin x
1
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 − 2
x cos axdx =
+
sin ax
2
a
a3
2
1
sin3 x
3
(73)
cos[(2a − b)x]
cos bx
cos ax sin bxdx =
−
4(2a − b)
2b
cos[(2a + b)x]
−
4(2a + b)
1
xn cosxdx = − (i)n+1 [Γ(n + 1, −ix)
2
+(−1)n Γ(n + 1, ix)]
2
1
cos3 ax
3a
eax cosh bxdx =
 ax
 e
 2
[a cosh bx − b sinh bx]
a − b2
2ax
x
e

+
4a
2
sinh axdx =
2
x2 cos xdx = 2x cos x + x2 − 2 sin x
(96)
(97)
eax tanh bxdx =
 (a+2b)x
e
a
a
2bx


 (a + 2b) 2 F1 1 + 2b , 1, 2 + 2b , −e


a
1
, 1, 1E, −e2bx
− eax 2 F1

a −1 ax 2b
 ax

 e − 2 tan [e ]


a
tanh ax dx =
sin[2(a − b)x]
x
sin 2ax
−
−
4
8a
16(a − b)
sin[2(a + b)x]
sin 2bx
+
−
(76)
8b
16(a + b)
1
(ia)1−n [(−1)n Γ(n + 1, −iax)
2
−Γ(n + 1, ixa)]
(98)
1
tan ax
a
(79)
(80)
1
1
ln cos ax +
sec2 ax
a
2a
sec xdx = ln | sec x + tan x| = 2 tanh−1 tan
1
tan ax
a
a=b
(113)
a=b
a = b (114)
a=b
1
ln cosh ax
a
(115)
1
[a sin ax cosh bx
a2 + b2
+b cos ax sinh bx]
(116)
cos ax cosh bxdx =
(100)
1
[b cos ax cosh bx+
a2 + b2
a sin ax sinh bx]
(117)
(101)
cos ax sinh bxdx =
(81)
x2 sin axdx =
2 2
1
[−a cos ax cosh bx+
a2 + b2
b sin ax sinh bx]
(118)
2−a x
2x sin ax
cos ax +
a3
a2
(102)
1
xn sin xdx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)]
2
(103)
1
[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
2
(82)
ex sin xdx =
1 x
e (sin x − cos x)
2
(83)
ebx sin axdx =
1
ebx (b sin ax − a cos ax)
a2 + b2
1
[−2ax + sinh 2ax]
4a
(120)
(104)
1
[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
2
tan3 axdx =
sin ax
x cos ax
+
a
a2
x sin axdx = −
(78)
tann axdx =
2 F1
(111)
a=b
(99)
(77)
1
tan axdx = − ln cos ax
a
tan2 axdx = −x +
x sin xdx = −x cos x + sin x
x2 sin xdx = 2 − x2 cos x + 2x sin x
sin 4ax
x
−
8
32a
a=b
xn cosaxdx =
sin2 ax cos2 bxdx =
sin2 ax cos2 axdx =
(110)
1
cosh ax
a
eax sinh bxdx =
 ax
 e
 2
[−b cosh bx + a sinh bx]
a − b2
2ax
x
e

−
4a
2
(74)
(75)
1
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 = −
1
ebx (a sin ax + b cos ax)
a2 + b2
cosh axdx =
3 sin ax
sin 3ax
+
4a
12a
sin2 x cos xdx =
ebx cos axdx =
Integrals of Hyperbolic Functions
1
csc axdx = − cot ax
a
(67)
1
cos1+p ax×
a(1 + p)
1+p 1 3+p
, ,
, cos2 ax
2
2
2
cos3 axdx =
x
= ln | csc x − cot x| + C
csc xdx = ln tan
2
2
sin 2ax
x
cos2 axdx = +
2
4a
2 F1
(106)
xex sin xdx =
(85)
(65)
3 cos ax
cos 3ax
+
4a
12a
cos axdx =
cosp axdx = −
1
sec2 x
2
n
sin3 axdx = −
1 x
e (sin x + cos x)
2
(64)
n
2 F1
ex cos xdx =
(63)
sec2 x tan xdx =
1
− cos ax
a
(84)
(105)