Table of Integrals

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 =
(19)
!
1
dx = 2 x ± a
x±a
(20)
"
1
dx = 2 a ! x
a! x
(21)
"x
(22)
!
(23)
! (ax + b)
1
1
! ax + b dx = a ln(ax + b)
1
"1
x+a
2
(x ! a)3/2
3
(18)
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 =
a
+ ln(a + x)
a+ x
ln(ax 2 + bx + c)
x
dx =
+ bx + c
2a
!!!!!"
©2005 BE Shapiro
(29)
"
# 2ax + b &
tan "1 %
$ 4ac " b 2 ('
a 4ac " b
4a
)
3/2
(
b 3 ln 2 a x + 2 b + ax
8a
)
5/2
(
)
(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 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
(
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
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.
1
(34)
"
a !x
(35)
!
x2 ± a2
(36)
"
a2 ! x2
(37)
!
(38)
"
2
x
x
x ±a
(40)
dx =
2
(
1
1
x x 2 ± a 2 ! ln x + x 2 ± a 2
2
2
)
# x a2 ! x2 &
1
1
dx = ! x a ! x 2 ! a 2 tan !1 % 2
(
2
2
2
a2 ! x2
$ x !a '
" b x%
ax 2 + bx + c
ax 2 + bx + c !dx = $
+
# 4a 2 '&
4ac ( b 2 " 2ax + b
%
!!!!!!!!!!!!!!+
ln $
+ 2 ax 2 + bc + c '
#
&
8a 3/2
a
" x ln(a
(51)
!e
(52)
!
b(4ac " b ) # 2ax + b
&
ln %
+ 2 ax 2 + bc + c (
$
'
16a 5/2
a
!!!!!!!!!!!!!!"
1#
b2 &
+ % x 2 " 2 ( ln(ax + b)
a '
2$
1
1#
a2 &
! b 2 x 2 )dx = ! x 2 + % x 2 ! 2 ( ln(a 2 ! bx 2 )
b '
2
2$
ax
dx =
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
$
'
1 ax
e
a
1
i "
xeax + 3/2 erf i ax
2a
a
(
xeax dx =
2
!
#
x
0
(54)
! xe
(55)
! x e dx = e (x
(56)
2
2x 2 &
2 ax
ax # x
x
e
dx
=
e
!
%$ a " a 2 + a 3 ('
(57)
! x e dx = e (x
(58)
!x e
2 x
x
3 x
x
n ax
!e
ax 2
x
#x 1&
dx = % " 2 ( eax
$a a '
2
3
dx = ( "1)
!(a, x) =
(59)
where
2
x
ax
)
e"t dt
! xe dx = (x " 1)e
!
(42)
2
2
(53)
ax 2 + bx + c !dx =
# x 3 bx 8ac " 3b 2 &
+
ax 2 + bx + c
!!!!!!!!!!!!!!! % +
24a 2 ('
$ 3 12a
!
(50)
erf (x) =
2
(41)
1
! x ln(ax + b)dx = 2a x " 4 x
EXPONENTIALS
x2
!x
b
(49)
dx = ! a 2 ! x 2
2
2
x
a
= x 2 ± a2
x
!
(39)
= sin !1
2
$
#
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
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
2
2
2
+ bx + c)dx =
©2005 BE Shapiro
! sin xdx = " cos x
(61)
! 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
(60)
(
)
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.
(67)
! sin
2
x cos xdx =
1
1
sin x " sin 3x
4
12
(68)
1
1
! sin x cos xdx = " 4 cos x " 12 cos 3x
(69)
2
2
! sin x cos xdx =
(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
!!!!!!!!!! (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
!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.
(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.