(1) This document provides formulas for integrals of common functions including polynomials, rational functions, radicals, and trigonometric functions. (2) The integrals are organized into sections based on the type of function, such as basic forms, integrals with roots, and rational functions. (3) Over 30 different integral formulas are provided in a table format with the integral given and its solution. This allows for easy reference to the antiderivatives of standard functions.

1. Table of Integrals
BASIC FORMS INTEGRALS WITH ROOTS
1 2
! x dx = n + 1 x " x ! adx = (x ! a)3/2
n n+1
(1) (18)
3
1 1
(2) ! x dx = ln x (19) ! x±a
dx = 2 x ± a
(3) ! udv = uv " ! vdu (20) "
1
dx = 2 a ! x
a! x
(4) " u(x)v!(x)dx = u(x)v(x) # " v(x)u !(x)dx 2 2
(21) "x x ! adx =
3
a(x ! a)3/2 + (x ! a)5/2
5
RATIONAL FUNCTIONS " 2b 2x %
(22) ! ax + bdx = $ +
# 3a 3 '&
b + ax
1 1
(5) ! ax + b dx = a ln(ax + b)
" 2b 2 4bx 2ax 2 %
! (ax + b) dx = b + ax $ + +
3/2
(23)
1 "1 # 5a 5 5 ' &
(6) ! (x + a) 2
dx =
x+a
x 2
(24) ! dx = ( x ± 2a ) x ± a
" a x % x±a 3
! (x + a) dx = (x + a) $ 1+n + 1+ n ' , n ! "1
n n
(7)
# &
x # x a! x&
(x + a)1+n (nx + x " a) (25) " a! x
dx = ! x a ! x ! a tan !1 %
$ x!a '
(
! x(x + a) dx = (n + 2)(n + 1)
n
(8)
x
(9)
dx
! 1+ x 2 = tan x
"1 (26) ! x+a
dx = x x + a " a ln # x + x + a %
$ &
# 4b 2 2bx 2x 2 &
(10)
dx 1 "1
! a 2 + x 2 = a tan (x / a) (27) !x ax + bdx = % " +
$ 15a 2 15a
+
5 (
'
b + ax
xdx 1 " b x x 3/2 %
(11) !a 2
= ln(a 2 + x 2 )
+ x2 2 ! x ax + bdx = $ +
2 '
b + ax
# 4a &
(28)
(12)
x 2 dx
! a 2 + x 2 = x " a tan (x / a)
"1
!!!!!!!!!!!!!!!!!!!!!!!!!(
(
b 2 ln 2 a x + 2 b + ax )
3/2
4a
x 3 dx 1 2 1 2
! a 2 + x 2 = 2 x " 2 a ln(a + x ) # b 2 x bx 3/2 x 5/2 &
2 2
(13)
!x ax + bdx = % " + + b + ax
3/2
$ 8a
2
12a 3 ( '
# 2ax + b & (29)
(14) " (ax
2
+ bx + c)!1 dx =
2
4ac ! b
tan !1 %
2
$ 4ac ! b 2 (
' "
(
b 3 ln 2 a x + 2 b + ax )
5/2
8a
( )
1 1
(15) ! (x + a)(x + b) dx = b " a [ ln(a + x) " ln(b + x)] , a ! b (30) ! x 2 ± a 2 dx =
1 1
x x 2 ± a 2 ± a 2 ln x + x 2 ± a 2
2 2
x a # x a2 ! x2 &
(16) ! (x + a) 2
dx =
a+ x
+ ln(a + x)
(31) " a 2 ! x 2 dx =
1 1
x a 2 ! x 2 ! a 2 tan !1 % 2 (
$ x !a '
2
2 2
x ln(ax 2 + bx + c)
! ax dx = 1
+ bx + c !x x 2 ± a 2 = (x 2 ± a 2 )3/2
2
2a (32)
(17) 3
b # 2ax + b &
tan "1 %
( )
!!!!!"
a 4ac " b $ 4ac " b 2 (
' 1
!
2
(33) dx = ln x + x 2 ± a 2
x ±a
2 2
©2005 BE Shapiro Page 1
This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or
suitability of this material for any purpose.

2. 1 x 1# b2 &
" = sin !1 b 1
(34)
! x ln(ax + b)dx = 2a x " 4 x + % x 2 " 2 ( ln(ax + b)
2
(49)
a !x
2 2 a 2$ a '
x 1# a2 &
(35) ! x2 ± a2
= x 2 ± a2 (50) " x ln(a
2 1
! b 2 x 2 )dx = ! x 2 + % x 2 ! 2 ( ln(a 2 ! bx 2 )
2 2$ b '
x
(36) " a2 ! x2
dx = ! a 2 ! x 2
EXPONENTIALS
( )
2
x 1 1 1 ax
(37) ! x ±a
2 2
dx =
2
x x 2 ± a 2 ! ln x + x 2 ± a 2
2
(51) !e
ax
dx =
a
e
i "
(38) "
x2 1 1 # x a2 ! x2 &
dx = ! x a ! x 2 ! a 2 tan !1 % 2 (
(52) ! xeax dx =
1
a
xeax + 3/2 erf i ax
2a
( ) where
$ x !a '
2
a2 ! x2 2 2
2 x
# e"t dt
2
erf (x) =
" b x% ! 0
! ax 2 + bx + c !dx = $ +
# 4a 2 '
&
ax 2 + bx + c
! xe dx = (x " 1)e
x x
(39) (53)
4ac ( b 2 " 2ax + b %
!!!!!!!!!!!!!!+ ln $ + 2 ax 2 + bc + c '
8a 3/2 # a & #x 1&
! xe dx = % " 2 ( eax
ax
(54)
$a a '
!x ax 2 + bx + c !dx =
! x e dx = e (x " 2x + 2)
2 x x 2
# x 3 bx 8ac " 3b 2 & (55)
(40) !!!!!!!!!!!!!!! % + + ax 2 + bx + c
$ 3 12a 24a 2 ( '
# x 2 2x 2 &
!!!!!!!!!!!!!!"
b(4ac " b ) # 2ax + b
ln %
2
&
+ 2 ax 2 + bc + c (
(56) ! x 2 eax dx = eax % " 2 + 3 (
$ a a a '
16a 5/2 $ a '
! x e dx = e (x " 3x 2 + 6x " 6)
3 x x 3
(57)
1 1 " 2ax + b %
(41) ! ax 2 + bx + c
dx = ln
a $ # a
+ 2 ax 2 + bx + c '
& 1
!x e dx = ( "1) #[1+ n, "ax] where
n ax n
(58)
x 1 a
! ax 2 + bx + c
dx =
a
ax 2 + bx + c
!(a, x) = $
#
t a"1e"t dt
x
(42)
b # 2ax + b &
+ 2 ax 2 + bx + c ( #
!!!!!" 3/2 ln %
2a $ a ' (59) !e
ax 2
dx = "i
2 a
erf ix a ( )
LOGARITHMS
(43) ! ln xdx = x ln x " x TRIGONOMETRIC FUNCTIONS
(44) !
ln(ax) 1
dx = ( ln(ax))
2 (60) ! sin xdx = " cos x
x 2
x 1
! sin xdx = " sin 2x
2
ax + b (61)
(45) ! ln(ax + b)dx = a
ln(ax + b) " x 2 4
3 1
! sin xdx = " cos x + cos 3x
3
2b "1 # ax & (62)
! ln(a x ± b )dx = x ln(a x ± b ) + a tan % b ( " 2x 4 12
2 2 2 2 2 2
(46)
$ '
2a !1 # bx &
(63) ! cos xdx = sin x
" ln(a ! b x )dx = x ln(a ! b x ) + b tan % a ( ! 2x
2 2 2 2 2 2
(47)
$ ' x 1
! cos xdx = + sin 2x
2
(64)
2 4
1 # 2ax + b &
! ln(ax + bx + c)dx = 4ac " b 2 tan "1 %
2
$ 4ac " b 2 (
3 1
' ! cos xdx = sin x + sin 3x
3
a (65)
(48) 4 12
# b &
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x + %
$ 2a
+ x ( ln ax 2 + bx + c
'
( ) 1
! sin x cos xdx = " 2 cos
2
(66) x
©2005 BE Shapiro Page 2
This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or
suitability of this material for any purpose.

3. ! sin
1 1
!x cos axdx =
n
(67) 2
x cos xdx = sin x " sin 3x
4 12 (89) 1
1 1 !!!!!!!!!! (ia)1"n $("1)n #(1+ n, "iax) " #(1+ n,iax) &
% '
! sin x cos xdx = " 4 cos x " 12 cos 3x 2
2
(68)
x 1
(90) ! x sin xdx = "x cos x + sin x
! sin x cos xdx = " sin 4 x
2 2
(69)
8 32 x 1
(91) ! x sin(ax)dx = " a cos ax + a 2
sin ax
(70) ! tan xdx = " ln cos x
!x sin xdx = (2 " x 2 )cos x + 2x sin x
2
(92)
! tan xdx = "x + tan x
2
(71)
2 " a2 x2 2
! x sin axdx =
3
1 (93) cos ax + 3 x sin ax
! tan xdx = ln[cos x] + sec 2 x a3 a
3
(72)
2
1
!x sin xdx = " (i)n $ #(n + 1, "ix) " ("1)n #(n + 1, "ix) &
n
(94) % '
(73) ! sec xdx = ln | sec x + tan x | 2
TRIGONOMETRIC FUNCTIONS WITH e ax
! sec xdx = tan x
2
(74)
1 x
(95) !e
x
sin xdx = e [ sin x " cos x ]
1 1
! sec xdx = 2 sec x tan x + 2 ln | sec x tan x |
(75) 3 2
1
(96) !e
bx
sin(ax)dx = ebx [ b sin ax " a cos ax ]
! sec x tan xdx = sec x b + a2
2
(76)
1 x
1 2 (97) !e
x
cos xdx = e [ sin x + cos x ]
! sec x tan xdx =
2
(77) sec x 2
2
1
1 n (98) !e
bx
cos(ax)dx = ebx [ a sin ax + b cos ax ]
! sec x tan xdx = sec x , n ! 0 b2 + a2
n
(78)
n
(79) ! csc xdx = ln | csc x " cot x | TRIGONOMETRIC FUNCTIONS WITH x n AND e ax
! csc xdx = " cot x
2
(80) 1 x
(99) ! xe
x
sin xdx = e [ cos x " x cos x + x sin x ]
2
1 1
! csc xdx = " cot x csc x + ln | csc x " cot x |
3
(81) 1 x
2 2 (100) ! xe
x
cos xdx = e [ x cos x " sin x + x sin x ]
2
1
! csc x cot xdx = " csc n x , n ! 0
n
(82)
n
HYPERBOLIC FUNCTIONS
(83) ! sec x csc xdx = ln tan x (101) ! cosh xdx = sinh x
eax
TRIGONOMETRIC FUNCTIONS WITH x n (102) !e
ax
cosh bxdx =
a " b2
2 [ a cosh bx " b sinh bx ]
(84) ! x cos xdx = cos x + x sin x (103) ! sinh xdx = cosh x
1 1
(85) ! x cos(ax)dx = a 2 cos ax + a x sin ax eax
[ "b cosh bx + a sinh bx ]
!e sinh bxdx =
ax
(104)
a " b2
2
!x cos xdx = 2x cos x + (x 2 " 2)sin x
2
(86)
!e tanh xdx = e x " 2 tan "1 (e x )
x
(105)
2 a2 x2 " 2
!x cos axdx = x cos ax +
2
(87) sin ax 1
! tanh axdx = a ln cosh ax
2
a a3 (106)
!x cos xdx =
n
(88) 1 1+n $
! cos ax cosh bxdx =
!!!!!!!!!" (i ) % #(1+ n, "ix) + ( "1)n #(1+ n,ix)&
'
(107) 1
2 !!!!!!!!!!
a + b2
2 [ a sin ax cosh bx + b cos ax sinh bx ]
©2005 BE Shapiro Page 3
This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or
suitability of this material for any purpose.

4. ! cos ax sinh bxdx =
(108) 1
!!!!!!!!!!
a + b2
2 [b cos ax cosh bx + a sin ax sinh bx ]
! sin ax cosh bxdx =
(109) 1
!!!!!!!!!!
a + b2
2 [ "a cos ax cosh bx + b sin ax sinh bx ]
! sin ax sinh bxdx =
(110) 1
!!!!!!!!!!
a + b2
2 [b cosh bx sin ax " a cos ax sinh bx ]
1
(111) ! sinh ax cosh axdx = 4a [ "2ax + sinh(2ax)]
! sinh ax cosh bxdx =
(112) 1
!!!!!!!!!!
b2 " a2
[b cosh bx sinh ax " a cosh ax sinh bx ]
©2005 BE Shapiro Page 4
This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or
suitability of this material for any purpose.