Table of Integrals
Table of Integrals
BASIC FORMS
(1)
! xndx = 1 xn+1 n+1
(2)
!
1 x
dx
=
ln
x
(3) ! udv = uv " ! vdu
(4) " u(x)v!(x)dx = u(x)v(x) # " v(x)u!(x)dx
RATIONAL FUNCTIONS
(5)
!
1 ax +
b
dx
=
1 a
ln(ax
+
b)
(6)
!
(x
1 + a)2
dx
=
"1 x+a
! (7)
(x
+
a)n
dx
=
(x
+
a)n
" #$
a 1+n
+
1
x +
n
% &'
,
n ! "1
! (8)
x(x + a)n dx = (x + a)1+n (nx + x " a)
(n + 2)(n + 1)
! (9)
dx 1+ x2
=
tan"1 x
! (10)
dx a2 + x2
=
1 tan"1(x / a) a
! (11)
xdx a2 + x2
=
1 ln(a2 2
+ x2 )
! (12)
x 2 dx a2 + x2
=
x " a tan"1(x / a)
! (13)
x 3 dx a2 + x2
=
1 x2 2
"
1 a2 ln(a2 2
+ x2 )
" (14)
(ax2 + bx + c)!1 dx =
2 4ac ! b2
tan!1
# $%
2ax + b & 4ac ! b2 '(
(15)
!
(
x
+
1 a)(x
+
b)
dx
=
b
1 "
a
[ln(a
+
x)
"
ln(b
+
x
)]
,
a!b
(16)
!
(x
x + a)2
dx
=
a
a +
x
+
ln(a
+
x)
! ax2
x + bx
+
c
dx
=
ln(ax2 + bx 2a
+
c)
(17)
!!!!!" a
b 4ac " b2
tan
"1
# $%
2ax + b & 4ac " b2 '(
INTEGRALS WITH ROOTS
" (18)
x ! adx = 2 (x ! a)3/2
3
(19)
!
1 dx = 2 x ? a x?a
(20)
"
1 dx = 2 a ! x a!x
" (21) x x ! adx = 2 a(x ! a)3/2 + 2 (x ! a)5/2
3
5
(22)
!
ax
+
bdx
=
" #$
2b 3a
+
2x 3
% &'
b + ax
! (23)
(ax + b)3/2 dx =
b+
ax
" #$
2b2 5a
+
4bx 5
+
2 ax 2 5
% &'
(24)
!
x dx = 2 (x ? 2a) x ? a
x?a 3
" (25)
a
x !
x
dx
=
!
x
a
!
x
!
a
tan!1
# $%
x a!x& x ! a '(
(26)
!
x dx = x+a
x
x + a " a ln #$ x +
x + a %&
! (27)
x
ax + bdx
=
# $%
"
4b2 15a2
+
2bx 15a
+
2x2 5
& '(
b + ax
!x
ax
+
bdx
=
" #$
bx 4a
+
x 3/2 2
% &'
b + ax
(28)
( ) b2 ln 2 a x + 2 b + ax
!!!!!!!!!!!!!!!!!!!!!!!!!(
4 a 3/2
! x3/2
ax + bdx
=
# $% "
b2 x 8a2
+
bx 3/2 12a
+
x5/2 3
& '(
b + ax
(29)
( ) b3 ln 2 a x + 2 b + ax
"
8a5/2
! ( ) (30)
x2 ? a2 dx = 1 x x2 ? a2 ? 1 a2 ln x + x2 ? a2
2
2
" (31)
a2 ! x2 dx = 1 x 2
a2
!
x2
!
1 2
a2
# tan!1 %
$
x
a2 ! x2 x2 ! a2
& ( '
! (32) x x2 ? a2 = 1 (x2 ? a2 )3/2 3
( ) ! (33)
1 dx = ln x + x2 ? a2
x2 ? a2
?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.
" (34)
1 = sin!1 x
a2 ! x2
a
! (35)
x
= x2 ? a2
x2 ? a2
" (36)
x dx = ! a2 ! x2
a2 ! x2
! ( ) (37)
x2 dx = 1 x x2 ? a2 ! 1 ln x + x2 ? a2
x2 ? a2
2
2
" (38)
x2 dx = ! 1 x
a2 ! x2
2
a!
x2
!
1 2
a2
# tan!1 %
$
x
a2 ! x2 x2 ! a2
& ( '
! ax2
+
bx
+
c
!dx
=
" #$
b 4a
+
x 2
% &'
ax2 + bx + c
(39)
!!!!!!!!!!!!!!+
4ac ( b2 8 a 3/2
ln
" #$
2ax + a
b
+
2
ax 2
+
bc
+
c
% &'
! x ax2 + bx + c !dx =
(40)
!!!!!!!!!!!!!!!#$%
x3 3
+
bx 12a
+
8ac " 3b2 24a2
& '(
ax2 + bx + c
!!!!!!!!!!!!!!"
b(4ac " b2 16a5/2
)
ln
# $%
2ax + a
b
+
2
ax 2
+
bc
+
c
& '(
! (41)
1
dx =
ax2 + bx + c
1 a
ln
" #$
2ax + a
b
+
2
ax 2
+
bx
+
c
% &'
!x
dx = 1 ax2 + bx + c
(42)
ax2 + bx + c a
!!!!!"
b 2 a 3/2
ln
# 2ax + $% a
b
+
2
ax 2
+
bx
+
c
& '(
LOGARITHMS
(43) ! ln xdx = x ln x " x
(44)
!
ln(ax) x
dx
=
1 2
(ln(ax))2
(45)
!
ln(ax
+
b)dx
=
ax + a
b
ln(ax
+
b)
"
x
! (46)
ln(a2 x2
?
b2
)dx
=
x
ln(a2
x2
?
b2
)
+
2b a
tan"1
# $%
ax & b '(
"
2x
" (47)
ln(a2
!
b2 x2
)dx
=
x ln(a2
!
b2 x2
)
+
2a b
tan!1
# $%
bx a
& '(
!
2x
! (48)
ln(ax2 + bx + c)dx = 1 a
4ac
"
b2
tan"1
# $%
2ax + b & 4ac " b2 '(
( ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"2
x
+
# $%
b 2a
+
x&'(
ln
ax 2
+ bx + c
! (49)
x ln(ax
+
b)dx
=
b 2a
x
"
1 4
x2
+
1 2
# $%
x2
"
b2 a2
& '(
ln(ax
+
b)
" (50)
x
ln(a2
!
b2 x2
)dx
=
!
1 2
x2
+
1 2
# $%
x2
!
a2 b2
& '(
ln(a2
!
bx 2
)
EXPONENTIALS
! (51) eaxdx = 1 eax a
! ( ) (52)
xeaxdx = 1 a
xeax
+
i" 2 a 3/2
erf
i
ax
where
# erf (x) = 2 x e"t2 dt
!0
! (53) xexdx = (x " 1)ex
! (54)
xeax
dx
=
# $%
x a
"
1 a2
& '(
eax
! (55) x2exdx = ex (x2 " 2x + 2)
! (56)
x2eax dx
=
eax
# $%
x2 a
"
2x a2
+
2 a3
& '(
! (57) x3exdx = ex (x3 " 3x2 + 6x " 6) ! (58) xneaxdx = ("1)n 1 #[1+ n, "ax] where
a
$ !(a, x) = # t a"1e"t dt x
( ) ! (59) eax2 dx = "i # erf ix a 2a
TRIGONOMETRIC FUNCTIONS
(60) ! sin xdx = " cos x
(61)
!
sin2
xdx
=
x 2
"
1 4
sin
2x
(62)
!
sin3
xdx
=
"
3 4
cos
x
+
1 12
cos
3x
(63) ! cos xdx = sin x
(64)
!
cos2
xdx
=
x 2
+
1 4
sin 2x
(65)
!
cos3
xdx
=
3 4
sin
x
+
1 12
sin 3x
(66)
!
sin
x
cos
xdx
=
"
1 2
cos2
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.
(67)
! sin2
x cos xdx
=
1 4
sin x
"
1 12
sin
3x
(68)
!
sin
x
cos2
xdx
=
"
1 4
cos
x
"
1 12
cos
3x
! (69) sin2 x cos2 xdx = x " 1 sin 4 x 8 32
(70) ! tan xdx = " ln cos x
(71) ! tan2 xdx = "x + tan x ! (72) tan3 xdx = ln[cos x] + 1 sec2 x
2
(73) ! sec xdx = ln | sec x + tan x |
(74) ! sec2 xdx = tan x
(75)
!
sec3
xdx
=
1 2
sec
x
tan
x
+
1 2
ln
|
sec
x
tan
x
|
(76) ! sec x tan xdx = sec x ! (77) sec2 x tan xdx = 1 sec2 x
2
! (78) secn x tan xdx = 1 secn x , n ! 0 n
! xn cos axdx =
(89)
!!!!!!!!!!
1 2
(ia)1"n
$%("1)n
#(1
+
n,
"iax)
"
#(1
+
n,
iax)&'
(90) ! x sin xdx = "x cos x + sin x
(91)
!
x
sin(ax)dx
=
"
x a
cos
ax
+
1 a2
sin
ax
! (92) x2 sin xdx = (2 " x2 )cos x + 2x sin x
! (93)
x3
sin axdx
=
2
" a2x2 a3
cos ax
+
2 a3
x sin ax
! (94)
xn
sin
xdx
=
"
1 2
(i)n
$%#(n
+
1,
"ix)
"
("1)n
#(n
+
1,
"ix)&'
TRIGONOMETRIC FUNCTIONS WITH eax
(95)
!
ex
sin
xdx
=
1 2
ex
[sin
x
"
cos
x]
! (96)
ebx
sin(ax)dx
=
b2
1 +
a2
ebx
[bsin ax
"
a cos ax]
(97)
!
ex
cos
xdx
=
1 2
ex
[sin
x
+
cos
x]
! (98)
ebx
cos(ax)dx
=
b2
1 +
a2
ebx
[ a sin
ax
+
b cos ax]
(79) ! csc xdx = ln | csc x " cot x |
(80) ! csc2 xdx = " cot x
(81)
!
csc3
xdx
=
"
1 2
cot
x
csc
x
+
1 2
ln
|
csc
x
"
cot
x
|
! (82) cscn x cot xdx = " 1 cscn x , n ! 0 n
(83) ! sec x csc xdx = ln tan x
TRIGONOMETRIC FUNCTIONS WITH xn AND eax
(99)
!
xex
sin
xdx
=
1 2
ex
[cos
x
"
x
cos
x
+
x sin
x]
(100)
!
xex
cos
xdx
=
1 2
ex
[x
cos
x
"
sin
x
+
x sin
x]
HYPERBOLIC FUNCTIONS
(101) ! cosh xdx = sinh x
TRIGONOMETRIC FUNCTIONS WITH xn
(84) ! x cos xdx = cos x + x sin x
(85)
!
x cos(ax)dx
=
1 a2
cos ax
+
1 a
x sin ax
! (86) x2 cos xdx = 2x cos x + (x2 " 2)sin x
! (87)
x2
cos axdx
=
2 a2
x cos ax
+
a2x2 " a3
2
sin ax
! xn cos xdx =
(88)
!!!!!!!!!"
1 2
(i
)1+n
$%#(1
+
n,
"ix)
+
("1)n
#(1
+
n,
ix)&'
?2005 BE Shapiro
! (102)
eax
cosh bxdx
=
eax a2 " b2
[a coshbx
"
b sinh bx ]
(103) ! sinh xdx = cosh x
! (104)
eax
sinh bxdx
=
eax a2 " b2
["b cosh bx
+
a sinh bx ]
! (105) ex tanh xdx = ex " 2 tan"1(ex )
(106)
!
tanh axdx
=
1 a
ln
cosh
ax
! cos ax cosh bxdx =
(107)
!!!!!!!!!!
a2
1 +
b2
[ a sin
ax
cosh bx
+
b cos ax sinh bx]
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.
! cos ax sinh bxdx =
(108)
!!!!!!!!!!
a2
1 +
b2
[b
cos
ax
cosh
bx
+
a
sin
ax
sinh
bx]
! sin ax cosh bxdx =
(109)
!!!!!!!!!!
a2
1 +
b2
["a
cos
ax
cosh bx
+
b sin
ax
sinh bx ]
! sin ax sinh bxdx =
(110)
!!!!!!!!!!
a2
1 +
b2
[b
cosh
bx
sin
ax
"
a
cos ax
sinh
bx]
(111)
! sinh ax cosh axdx =
1 4a
["2ax
+
sinh(2ax
)]
! sinh ax cosh bxdx =
(112)
!!!!!!!!!! b2
1 "
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.
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