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
n+1
1
RATIONAL FUNCTIONS
(5)
" x ! adx = 3 (x ! a)
(19)
! x ± a dx = 2 x ± a
(20)
" a ! x dx = 2 a ! x
(21)
" x x ! adx = 3 a(x ! a)
(22)
! ax + bdx = $# 3a + 3 '& b + ax
(23)
! (ax + b) dx = b + ax $# 5a + 5 + 5 '&
1
1
! ax + b dx = a ln(ax + b)
"1
1
2
(18)
3/2
1
1
2
" 2b
2
+ (x ! a)5/2
5
3/2
2x %
" 2b 2
3/2
4bx
2ax 2 %
(6)
! (x + a) dx = x + a
(24)
(7)
x %
" a
! (x + a) dx = (x + a) $# 1+n + 1+ n '& , n ! "1
! x ± a dx = 3 ( x ± 2a ) x ± a
(25)
(8)
(x + a)1+n (nx + x " a)
! x(x + a) dx = (n + 2)(n + 1)
" a ! x dx = ! x a ! x ! a tan %$
(9)
dx
"1
! 1+ x 2 = tan x
(26)
! x + a dx = x x + a " a ln #$ x + x + a %&
(10)
1 "1
dx
! a 2 + x 2 = a tan (x / a)
(27)
! x ax + bdx = %$ " 15a + 15a + 5 (' b + ax
(11)
! a + x = 2 ln(a + x )
(12)
x 2 dx
"1
! a 2 + x 2 = x " a tan (x / a)
2
n
n
n
1
xdx
2
2
(28)
(13)
(14)
" (ax + bx + c) dx = 4ac ! b tan %$ 4ac ! b ('
2
!1
# 2ax + b &
(15)
(16)
! (x + a) dx = a + x + ln(a + x)
x
2
(17)
ln(ax 2 + bx + c)
2a
!!!!!"
©2005 BE Shapiro
4b 2
# 2ax + b &
tan "1 %
$ 4ac " b 2 ('
a 4ac " b
x 3/2 %
! x ax + bdx = $# 4a + 2 '& b + ax
(
b 2 ln 2 a x + 2 b + ax
4a
)
3/2
# b 2 x bx 3/2 x 5/2 &
b + ax
ax + bdx = % "
+
+
2
12a
3 ('
$ 8a
3/2
"
(
b 3 ln 2 a x + 2 b + ax
8a
)
5/2
(30)
! x ± a dx = 2 x x ± a ± 2 a ln ( x + x ± a )
(31)
" a ! x dx = 2 x a ! x ! 2 a tan %$ x ! a
(32)
! x x ± a = 3 (x ± a )
(33)
! x ± a dx = ln ( x + x ± a )
b
2
2x 2 &
2bx
2
(29)
a
2
! ax + bx + c dx =
#
2
1
1
! (x + a)(x + b) dx = b " a [ ln(a + x) " ln(b + x)] , a ! b
# x a! x&
x ! a ('
x
!x
2
x
!1
!!!!!!!!!!!!!!!!!!!!!!!!!(
1 2 1 2
x 3 dx
2
2
! a 2 + x 2 = 2 x " 2 a ln(a + x )
!1
x
"b x
2
2
2
2
x
2
2
2
2
2
2
1
2
1
2
1
2
1
2
2
2
1
2
1
2
2
!1
2
# x a2 ! x2 &
(
2
2
'
2 3/2
2
2
2
Page 1
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1
x
(34)
" a ! x = sin a
(35)
! x ±a = x ±a
!1
2
2
x
2
2
x
(36)
" a ! x dx = ! a ! x
(37)
! x ± a dx = 2 x x ± a ! 2 ln ( x + x ± a )
(38)
" a ! x dx = ! 2 x a ! x ! 2 a tan %$ x ! a
2
2
" x ln(a ! b x )dx = ! 2 x + 2 %$ x ! b (' ln(a ! bx )
x
2
1
2
2
1
2
1
2
2
1
2
!1
2
2
2
" b
# x a2 ! x2 &
(
2
2
'
2
1
2
2
1# 2
a2 &
x%
2
1 ax
(51)
! e dx = a e
(52)
! xe dx = a xe + 2a
ax
1
ax
2
i "
ax
3/2
(
erf i ax
) where
2 x "t 2
e dt
! #0
(53)
! xe dx = (x " 1)e
(54)
! xe dx = %$ a " a (' e
(55)
! x e dx = e (x " 2x + 2)
b(4ac " b ) # 2ax + b
&
ln %
+ 2 ax 2 + bc + c (
$
'
16a 5/2
a
(56)
2
2x 2 &
2 ax
ax # x
x
e
dx
=
e
!
%$ a " a 2 + a 3 ('
1 " 2ax + b
%
dx =
ln
+ 2 ax 2 + bx + c '
a $#
a
&
ax 2 + bx + c
(57)
! x e dx = e (x " 3x + 6x " 6)
(58)
! x e dx = ( "1) a #[1+ n, "ax] where
4ac ( b 2 " 2ax + b
%
!!!!!!!!!!!!!!+
ln $
+ 2 ax 2 + bc + c '
#
&
8a 3/2
a
# x 3 bx 8ac " 3b 2 &
+
ax 2 + bx + c
!!!!!!!!!!!!!!! % +
24a 2 ('
$ 3 12a
1
1
x
! ax + bx + c dx = a ax + bx + c
2
x
ax
#x
x
1 & ax
2
2 x
x
3 x
2
x
3
2
n
n ax
1
#
!(a, x) = $ t a"1e"t dt
2
!!!!!"
2
2
erf (x) =
! ax + bx + c !dx = $# 4a + 2 '& ax + bx + c
!!!!!!!!!!!!!!"
(42)
2
EXPONENTIALS
2
x2
!
2
2
2
(41)
b2 &
(50)
2
! x ax + bx + c !dx =
(40)
1# 2
2
2
2
(39)
1
! x ln(ax + b)dx = 2a x " 4 x + 2 $% x " a '( ln(ax + b)
2
2
b
(49)
x
b
# 2ax + b
&
ln
+ 2 ax 2 + bx + c (
2a 3/2 %$
a
'
(59)
#
! e dx = "i 2 a erf (ix a )
ax 2
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
TRIGONOMETRIC FUNCTIONS
(60)
! sin xdx = " cos x
(61)
! sin xdx = 2 " 4 sin 2x
(62)
! sin xdx = " 4 cos x + 12 cos 3x
(63)
! cos xdx = sin x
(64)
! cos xdx = 2 + 4 sin 2x
! ln(ax + bx + c)dx = a 4ac " b tan %$ 4ac " b ('
(65)
# b
&
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x + %
+ x ( ln ax 2 + bx + c
$ 2a
'
! cos xdx = 4 sin x + 12 sin 3x
(66)
! sin x cos xdx = " 2 cos x
1
ln(ax)
2
dx = ( ln(ax))
2
x
2
2
2
ax + b
ln(ax + b) " x
a
2
2
2
2
2
2
1
2
2
2
2
2
"1
# 2ax + b &
2
(48)
©2005 BE Shapiro
(
)
2
x
3
3
2
3
1
x
1
1
3
1
1
2
Page 2
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suitability of this material for any purpose.
1
1
(67)
! sin x cos xdx = 4 sin x " 12 sin 3x
(68)
1
1
! sin x cos xdx = " 4 cos x " 12 cos 3x
(69)
2
2
! sin x cos xdx =
2
n
(89)
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 sin ax
(92)
! x sin xdx = (2 " x )cos x + 2x sin x
(93)
3
! x sin axdx =
! x sin xdx = " 2 (i) $% #(n + 1, "ix) " ("1) #(n + 1, "ix)&'
2
x 1
" sin 4 x
8 32
(70)
! tan xdx = " ln cos x
(71)
2
! tan xdx = "x + tan x
1
! x cos axdx =
x
1
2
2
2
2 " a2 x2
2
cos ax + 3 x sin ax
a3
a
(72)
! tan xdx = ln[cos x] + 2 sec x
(73)
! sec xdx = ln | sec x + tan x |
(94)
(74)
! sec xdx = tan x
TRIGONOMETRIC FUNCTIONS WITH e ax
(75)
1
1
! sec xdx = 2 sec x tan x + 2 ln | sec x tan x |
3
2
2
1
n
n
n
1 x
(95)
! e sin xdx = 2 e [sin x " cos x ]
(96)
! e sin(ax)dx = b + a e [b sin ax " a cos ax ]
(97)
! e cos xdx = 2 e [sin x + cos x ]
(98)
! e cos(ax)dx = b + a e [ a sin ax + b cos ax ]
3
x
1
bx
bx
(76)
! sec x tan xdx = sec x
(77)
! sec x tan xdx = 2 sec x
(78)
! sec x tan xdx = n sec x , n ! 0
(79)
! csc xdx = ln | csc x " cot x |
TRIGONOMETRIC FUNCTIONS WITH x n AND e ax
(80)
! csc xdx = " cot x
(99)
(81)
! csc xdx = " 2 cot x csc x + 2 ln | csc x " cot x |
(82)
! csc x cot xdx = " n csc x , n ! 0
(83)
! sec x csc xdx = ln tan x
2
n
1
2
1
n
2
1
3
1
1
n
1
bx
2
bx
2
1 x
! xe sin xdx = 2 e [ cos x " x cos x + x sin x ]
x
(100) ! xe x cos xdx =
1 x
e [ x cos x " sin x + x sin x ]
2
(101) ! cosh xdx = sinh x
! x cos xdx = cos x + x sin x
(85)
1
1
! x cos(ax)dx = a 2 cos ax + a x sin ax
(102) ! eax cosh bxdx =
eax
[ a cosh bx " b sinh bx ]
a " b2
2
(103) ! sinh xdx = cosh x
(86)
! x cos xdx = 2x cos x + (x " 2)sin x
(87)
! x cos axdx = a x cos ax +
(104) ! eax sinh bxdx =
eax
[ "b cosh bx + a sinh bx ]
a " b2
2
2
2
2
2
a2 x2 " 2
sin ax
a3
(105) ! e x tanh xdx = e x " 2 tan "1 (e x )
(106) ! tanh axdx =
! x cos xdx =
n
!!!!!!!!!"
1 x
x
HYPERBOLIC FUNCTIONS
(84)
(88)
2
n
TRIGONOMETRIC FUNCTIONS WITH x n
2
2
1 1+n $
(i ) % #(1+ n, "ix) + ( "1)n #(1+ n,ix)&'
2
©2005 BE Shapiro
(107)
1
ln cosh ax
a
! cos ax cosh bxdx =
1
!!!!!!!!!! 2
[ a sin ax cosh bx + b cos ax sinh bx ]
a + b2
Page 3
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suitability of this material for any purpose.
(108)
(109)
(110)
! cos ax sinh bxdx =
1
!!!!!!!!!! 2
[b cos ax cosh bx + a sin ax sinh bx ]
a + b2
! sin ax cosh bxdx =
1
!!!!!!!!!! 2
[ "a cos ax cosh bx + b sin ax sinh bx ]
a + b2
! sin ax sinh bxdx =
1
!!!!!!!!!! 2
[b cosh bx sin ax " a cos ax sinh bx ]
a + b2
(111) ! sinh ax cosh axdx =
(112)
1
[ "2ax + sinh(2ax)]
4a
! sinh ax cosh bxdx =
1
!!!!!!!!!! 2
[b cosh bx sinh ax " a cosh ax sinh bx ]
b " a2
©2005 BE Shapiro
Page 4
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suitability of this material for any purpose.