This document provides formulas for integrals of common functions. It includes integrals involving roots, rational functions, exponentials, logarithms, trigonometric functions, hyperbolic functions, and combinations of these functions with exponents. Some example integrals listed are the integral of x from 1 to n, the integral of secant cubed x, and the integral of sine of ax times cosine of bx. Over 30 integrals are listed in the document.

1. Table of Integrals
BASIC FORMS
INTEGRALS WITH ROOTS
1
(1)
! x dx = n + 1 x
(2)
! x dx = ln x
(3)
! udv = uv " ! vdu
(4)
" u(x)v!(x)dx = u(x)v(x) # " v(x)u !(x)dx
n
"
x ! adx =
(19)
!
1
dx = 2 x ± a
x±a
(20)
"
1
dx = 2 a ! x
a! x
(21)
"x
(22)
!
(23)
1
RATIONAL FUNCTIONS
(5)
! (ax + b)
1
1
! ax + b dx = a ln(ax + b)
1
2
(x ! a)3/2
3
(18)
n+1
"1
x+a
x ! adx =
2
2
a(x ! a)3/2 + (x ! a)5/2
3
5
" 2b 2x %
b + ax
ax + bdx = $
+
# 3a 3 '
&
3/2
" 2b 2 4bx 2ax 2 %
dx = b + ax $
+
+
5 '
5
# 5a
&
(6)
! (x + a)
(24)
(7)
x %
" a
! (x + a) dx = (x + a) $ 1+n + 1+ n ' , n ! "1
#
&
!
2
x
dx = ( x ± 2a ) x ± a
3
x±a
(25)
(8)
(x + a)1+n (nx + x " a)
! x(x + a) dx = (n + 2)(n + 1)
"
# x a! x&
x
dx = ! x a ! x ! a tan !1 %
(
a! x
$ x!a '
(9)
dx
"1
! 1+ x 2 = tan x
(26)
!
x
dx = x x + a " a ln # x + x + a %
$
&
x+a
(10)
1 "1
dx
! a 2 + x 2 = a tan (x / a)
(27)
!x
(11)
!a
(12)
x 2 dx
"1
! a 2 + x 2 = x " a tan (x / a)
2
dx =
n
n
n
1
xdx
= ln(a 2 + x 2 )
+ x2 2
!
2
(28)
(13)
(14)
" (ax
+ bx + c)!1 dx =
!x
# 2ax + b &
tan !1 %
$ 4ac ! b 2 (
'
4ac ! b
2
(15)
(16)
! (x + a)
(17)
! ax
2
2
dx =
ln(ax 2 + bx + c)
x
dx =
+ bx + c
2a
©2005 BE Shapiro
4a
)
3/2
(29)
"
(
b 3 ln 2 a x + 2 b + ax
8a
)
5/2
(
)
# 2ax + b &
tan "1 %
$ 4ac " b 2 (
'
a 4ac " b
(30)
!
x 2 ± a 2 dx =
1
1
x x 2 ± a 2 ± a 2 ln x + x 2 ± a 2
2
2
(31)
a
+ ln(a + x)
a+ x
!!!!!"
(
b 2 ln 2 a x + 2 b + ax
# b 2 x bx 3/2 x 5/2 &
b + ax
ax + bdx = % "
+
+
2
12a
3 (
$ 8a
'
3/2
2
1
1
! (x + a)(x + b) dx = b " a [ ln(a + x) " ln(b + x)] , a ! b
x
" b x x 3/2 %
b + ax
x ax + bdx = $
+
2 '
# 4a
&
!!!!!!!!!!!!!!!!!!!!!!!!!(
1 2 1 2
x 3 dx
2
2
! a 2 + x 2 = 2 x " 2 a ln(a + x )
2
# 4b 2 2bx 2x 2 &
ax + bdx = % "
+
+
b + ax
5 (
$ 15a 2 15a
'
"
a 2 ! x 2 dx =
# x a2 ! x2 &
1
1
x a 2 ! x 2 ! a 2 tan !1 % 2
(
2
2
2
$ x !a '
(32)
!x
(33)
!
b
2
1
x 2 ± a 2 = (x 2 ± a 2 )3/2
3
1
x ±a
2
2
(
dx = ln x + x 2 ± a 2
)
Page 1
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suitability of this material for any purpose.

2. 1
(34)
"
a !x
(35)
!
x2 ± a2
(36)
"
a2 ! x2
(37)
x
x ±a
(40)
dx =
2
(
1
1
x x 2 ± a 2 ! ln x + x 2 ± a 2
2
2
)
(51)
" b x%
ax 2 + bx + c
ax 2 + bx + c !dx = $
+
# 4a 2 '
&
!e
(52)
!
1
1#
a2 &
! b 2 x 2 )dx = ! x 2 + % x 2 ! 2 ( ln(a 2 ! bx 2 )
b '
2
2$
ax
dx =
1 ax
e
a
1
i "
xeax + 3/2 erf i ax
2a
a
(
xeax dx =
2
!
#
x
0
! xe
(55)
! x e dx = e (x
(56)
b(4ac " b ) # 2ax + b
&
ln %
+ 2 ax 2 + bc + c (
$
'
16a 5/2
a
1 " 2ax + b
%
dx =
ln
+ 2 ax 2 + bx + c '
a $
a
#
&
ax 2 + bx + c
1
1
x
dx =
ax 2 + bx + c
a
ax 2 + bx + c
b
# 2ax + b
&
!!!!!" 3/2 ln %
+ 2 ax 2 + bx + c (
2a
a
$
'
where
2
(54)
# x 3 bx 8ac " 3b 2 &
+
ax 2 + bx + c
!!!!!!!!!!!!!!! % +
24a 2 (
$ 3 12a
'
)
e"t dt
! xe dx = (x " 1)e
# x 2 2x 2 &
x 2 eax dx = eax % " 2 + 3 (
!
a '
$ a a
(57)
! x e dx = e (x
(58)
!x e
ax 2 + bx + c !dx =
!!!!!!!!!!!!!!"
!
(42)
1#
b2 &
+ % x 2 " 2 ( ln(ax + b)
a '
2$
(53)
4ac ( b 2 " 2ax + b
%
!!!!!!!!!!!!!!+
ln $
+ 2 ax 2 + bc + c '
#
&
8a 3/2
a
!
2
erf (x) =
2
(41)
" x ln(a
2
EXPONENTIALS
# x a2 ! x2 &
1
1
dx = ! x a ! x 2 ! a 2 tan !1 % 2
(
2
2
2
a2 ! x2
$ x !a '
!x
1
! x ln(ax + b)dx = 2a x " 4 x
(50)
x2
!
b
(49)
dx = ! a 2 ! x 2
2
2
x
a
= x 2 ± a2
x
"
(39)
= sin !1
2
x
!
(38)
2
x
ax
#x 1&
dx = % " 2 ( eax
$a a '
2 x
x
3 x
x
n ax
!e
ax 2
2
3
dx = ( "1)
!(a, x) =
(59)
x
$
#
x
dx = "i
n
" 2x + 2)
" 3x 2 + 6x " 6)
1
#[1+ n, "ax] where
a
t a"1e"t dt
#
erf ix a
2 a
(
)
LOGARITHMS
(43)
! ln xdx = x ln x " x
(44)
!
(45)
! ln(ax + b)dx =
(46)
2b "1 # ax &
! ln(a x ± b )dx = x ln(a x ± b ) + a tan % b ( " 2x
$ '
(47)
2a !1 # bx &
" ln(a ! b x )dx = x ln(a ! b x ) + b tan % a ( ! 2x
$ '
(48)
TRIGONOMETRIC FUNCTIONS
(60)
2
2
2
ax + b
ln(ax + b) " x
a
2
2
2
2
2
2
2
2
2
+ bx + c)dx =
©2005 BE Shapiro
! sin
(62)
! sin
(63)
! cos xdx = sin x
(64)
! cos
(65)
! cos
(66)
! sin x cos xdx = " 2 cos
2
# 2ax + b &
1
4ac " b 2 tan "1 %
a
$ 4ac " b 2 (
'
# b
&
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x + %
+ x ( ln ax 2 + bx + c
$ 2a
'
! ln(ax
! sin xdx = " cos x
(61)
1
ln(ax)
2
dx = ( ln(ax))
2
x
(
)
2
3
xdx =
x 1
" sin 2x
2 4
3
1
xdx = " cos x + cos 3x
4
12
2
xdx =
x 1
+ sin 2x
2 4
3
xdx =
3
1
sin x + sin 3x
4
12
1
2
x
Page 2
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suitability of this material for any purpose.

3. (67)
! sin
2
x cos xdx =
1
1
sin x " sin 3x
4
12
(68)
2
2
! sin x cos xdx =
1
!!!!!!!!!! (ia)1"n $("1)n #(1+ n, "iax) " #(1+ n,iax) &
%
'
2
(90)
! x sin xdx = "x cos x + sin x
(91)
! x sin(ax)dx = " a cos ax + a
(92)
!x
(93)
3
! x sin axdx =
!x
2
x 1
" sin 4 x
8 32
(70)
! tan xdx = " ln cos x
(71)
2
! tan xdx = "x + tan x
1
xdx = ln[cos x] + sec 2 x
2
cos axdx =
n
(89)
1
1
! sin x cos xdx = " 4 cos x " 12 cos 3x
(69)
!x
x
1
2
sin ax
sin xdx = (2 " x 2 )cos x + 2x sin x
2
2 " a2 x2
2
cos ax + 3 x sin ax
a3
a
(72)
! tan
(73)
! sec xdx = ln | sec x + tan x |
(94)
(74)
! sec
TRIGONOMETRIC FUNCTIONS WITH e ax
(75)
1
1
! sec xdx = 2 sec x tan x + 2 ln | sec x tan x |
3
2
xdx = tan x
(95)
!e
(96)
!e
3
1
sin xdx = " (i)n $ #(n + 1, "ix) " ("1)n #(n + 1, "ix) &
%
'
2
n
x
sin xdx =
1 x
e [ sin x " cos x ]
2
sin(ax)dx =
bx
1
ebx [ b sin ax " a cos ax ]
b + a2
(76)
! sec x tan xdx = sec x
(77)
! sec
(78)
! sec
(79)
! csc xdx = ln | csc x " cot x |
TRIGONOMETRIC FUNCTIONS WITH x n AND e ax
(80)
! csc
(99)
! xe
(81)
! csc
(100)
! xe
(82)
! csc
(83)
! sec x csc xdx = ln tan x
2
x tan xdx =
1 2
sec x
2
(97)
!e
n
x tan xdx =
1 n
sec x , n ! 0
n
(98)
!e
2
3
n
xdx = " cot x
1
1
xdx = " cot x csc x + ln | csc x " cot x |
2
2
x
1 x
e [ sin x + cos x ]
2
cos xdx =
cos(ax)dx =
bx
1
ebx [ a sin ax + b cos ax ]
b2 + a2
x
sin xdx =
1 x
e [ cos x " x cos x + x sin x ]
2
x
cos xdx =
1 x
e [ x cos x " sin x + x sin x ]
2
1
x cot xdx = " csc n x , n ! 0
n
TRIGONOMETRIC FUNCTIONS WITH x n
2
HYPERBOLIC FUNCTIONS
(101)
! cosh xdx = sinh x
(102)
!e
ax
cosh bxdx =
eax
[ a cosh bx " b sinh bx ]
a " b2
2
(84)
! x cos xdx = cos x + x sin x
(103)
! sinh xdx = cosh x
(85)
1
1
! x cos(ax)dx = a 2 cos ax + a x sin ax
(104)
!e
(86)
!x
(105)
!e
(87)
!x
(106)
! tanh axdx = a ln cosh ax
(88)
!x
2
2
n
cos xdx = 2x cos x + (x 2 " 2)sin x
cos axdx =
2
a2 x2 " 2
x cos ax +
sin ax
2
a
a3
cos xdx =
!!!!!!!!!"
1 1+n $
(i ) % #(1+ n, "ix) + ( "1)n #(1+ n,ix)&
'
2
©2005 BE Shapiro
(107)
ax
x
sinh bxdx =
eax
[ "b cosh bx + a sinh bx ]
a " b2
2
tanh xdx = e x " 2 tan "1 (e x )
1
! cos ax cosh bxdx =
!!!!!!!!!!
1
[ a sin ax cosh bx + b cos ax sinh bx ]
a + b2
2
Page 3
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suitability of this material for any purpose.

4. (108)
! cos ax sinh bxdx =
!!!!!!!!!!
(109)
! sin ax cosh bxdx =
!!!!!!!!!!
(110)
(112)
1
[ "a cos ax cosh bx + b sin ax sinh bx ]
a + b2
2
! sin ax sinh bxdx =
!!!!!!!!!!
(111)
1
[b cos ax cosh bx + a sin ax sinh bx ]
a + b2
2
1
[b cosh bx sin ax " a cos ax sinh bx ]
a + b2
2
1
! sinh ax cosh axdx = 4a [ "2ax + sinh(2ax)]
! sinh ax cosh bxdx =
!!!!!!!!!!
1
[b cosh bx sinh ax " a cosh ax sinh bx ]
b2 " a2
©2005 BE Shapiro
Page 4
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suitability of this material for any purpose.