Integral table

integrals

1. Table of Basic Integrals
Basic Forms
(1)
Z
x
n
dx=
1
n+ 1
x
n+1
; n6=1
(2)
Z
1
x
dx= lnjxj
(3)
Z
udv=uv
Z
vdu
(4)
Z
1
ax+b
dx=
1
a
lnjax+bj
Integrals of Rational Functions
(5)
Z
1
(x+a)
2
dx=
1
x+a
(6)
Z
(x+a)
n
dx=
(x+a)
n+1
n+ 1
; n6=1
(7)
Z
x(x+a)
n
dx=
(x+a)
n+1
((n+ 1)xa)
(n+ 1)(n+ 2)
(8)
Z
1
1 +x
2
dx= tan
1
x
(9)
Z
1
a
2
+x
2
dx=
1
a
tan
1
x
a
1

2. (10)
Z
x
a
2
+x
2
dx=
1
2
lnja
2
+x
2
j
(11)
Z
x
2
a
2
+x
2
dx=xatan
1
x
a
(12)
Z
x
3
a
2
+x
2
dx=
1
2
x
2

1
2
a
2
lnja
2
+x
2
j
(13)
Z
1
ax
2
+bx+c
dx=
2
p
4acb
2
tan
1
2ax+b
p
4acb
2
(14)
Z
1
(x+a)(x+b)
dx=
1
ba
ln
a+x
b+x
; a6=b
(15)
Z
x
(x+a)
2
dx=
a
a+x
+ lnja+xj
(16)
Z
x
ax
2
+bx+c
dx=
1
2a
lnjax
2
+bx+cj
b
a
p
4acb
2
tan
1
2ax+b
p
4acb
2
Integrals with Roots
(17)
Z
p
xa dx=
2
3
(xa)
3=2
(18)
Z
1
p
xa
dx= 2
p
xa
(19)
Z
1
p
ax
dx=2
p
ax
2

3. (20)
Z
x
p
xa dx=
8
<
:
2a
3
(xa)
3=2
+
2
5
(xa)
5=2
;or
2
3
x(xa)
3=2

4
15
(xa)
5=2
;or
2
15
(2a+ 3x)(xa)
3=2
(21)
Z
p
ax+b dx=

2b
3a
+
2x
3

p
ax+b
(22)
Z
(ax+b)
3=2
dx=
2
5a
(ax+b)
5=2
(23)
Z
x
p
xa
dx=
2
3
(x2a)
p
xa
(24)
Zr
x
ax
dx=
p
x(ax)atan
1
p
x(ax)
xa
(25)
Zr
x
a+x
dx=
p
x(a+x)aln
p
x+
p
x+a

(26)
Z
x
p
ax+b dx=
2
15a
2
(2b
2
+abx+ 3a
2
x
2
)
p
ax+b
(27)
Z
p
x(ax+b)dx=
1
4a
3=2
h
(2ax+b)
p
ax(ax+b)b
2
ln


a
p
x+
p
a(ax+b)



i
(28)
Z
p
x
3
(ax+b)dx=

b
12a

b
2
8a
2
x
+
x
3

p
x
3
(ax+b)+
b
3
8a
5=2
ln


a
p
x+
p
a(ax+b)



(29)
Z
p
x
2
a
2
dx=
1
2
x
p
x
2
a
2

1
2
a
2
ln


x+
p
x
2
a
2



3

4. (30)
Z
p
a
2
x
2
dx=
1
2
x
p
a
2
x
2
+
1
2
a
2
tan
1
x
p
a
2
x
2
(31)
Z
x
p
x
2
a
2
dx=
1
3

x
2
a
2

3=2
(32)
Z
1
p
x
2
a
2
dx= ln


x+
p
x
2
a
2



(33)
Z
1
p
a
2
x
2
dx= sin
1
x
a
(34)
Z
x
p
x
2
a
2
dx=
p
x
2
a
2
(35)
Z
x
p
a
2
x
2
dx=
p
a
2
x
2
(36)
Z
x
2
p
x
2
a
2
dx=
1
2
x
p
x
2
a
2

1
2
a
2
ln


x+
p
x
2
a
2



(37)
Z
p
ax
2
+bx+c dx=
b+ 2ax
4a
p
ax
2
+bx+c+
4acb
2
8a
3=2
ln


2ax+b+ 2
p
a(ax
2
+bx
+
c)



Z
x
p
ax
2
+bx+c dx=
1
48a
5=2

2
p
a
p
ax
2
+bx+c

3b
2
+ 2abx+ 8a(c+ax
2
)

+3(b
3
4abc) ln


b+ 2ax+ 2
p
a
p
ax
2
+bx+c




(38)
4

5. (39)
Z
1
p
ax
2
+bx+c
dx=
1
p
a
ln


2ax+b+ 2
p
a(ax
2
+bx+c)



(40)
Z
x
p
ax
2
+bx+c
dx=
1
a
p
ax
2
+bx+c
b
2a
3=2
ln


2ax+b+ 2
p
a(ax
2
+bx+c)



(41)
Z
dx
(a
2
+x
2
)
3=2
=
x
a
2
p
a
2
+x
2
Integrals with Logarithms
(42)
Z
lnax dx=xlnaxx
(43)
Z
xlnx dx=
1
2
x
2
lnx
x
2
4
(44)
Z
x
2
lnx dx=
1
3
x
3
lnx
x
3
9
(45)
Z
x
n
lnx dx=x
n+1

lnx
n+ 1

1
(n+ 1)
2

; n6=1
(46)
Z
lnax
x
dx=
1
2
(lnax)
2
(47)
Z
lnx
x
2
dx=
1
x

lnx
x
5

6. (48)
Z
ln(ax+b)dx=

x+
b
a

ln(ax+b)x; a6= 0
(49)
Z
ln(x
2
+a
2
)dx=xln(x
2
+a
2
) + 2atan
1
x
a
2x
(50)
Z
ln(x
2
a
2
)dx=xln(x
2
a
2
) +aln
x+a
xa
2x
(51)
Z
ln

ax
2
+bx+c

dx=
1
a
p
4acb
2
tan
1
2ax+b
p
4acb
2
2x+

b
2a
+x

ln

ax
2
+bx+c

(52)
Z
xln(ax+b)dx=
bx
2a

1
4
x
2
+
1
2

x
2

b
2
a
2

ln(ax+b)
(53)
Z
xln

a
2
b
2
x
2

dx=
1
2
x
2
+
1
2

x
2

a
2
b
2

ln

a
2
b
2
x
2

(54)
Z
(lnx)
2
dx= 2x2xlnx+x(lnx)
2
(55)
Z
(lnx)
3
dx=6x+x(lnx)
3
3x(lnx)
2
+ 6xlnx
(56)
Z
x(lnx)
2
dx=
x
2
4
+
1
2
x
2
(lnx)
2

1
2
x
2
lnx
(57)
Z
x
2
(lnx)
2
dx=
2x
3
27
+
1
3
x
3
(lnx)
2

2
9
x
3
lnx
6

7. Integrals with Exponentials
(58)
Z
e
ax
dx=
1
a
e
ax
(59)
Z
p
xe
ax
dx=
1
a
p
xe
ax
+
i
p

2a
3=2
erf

i
p
ax

;where erf(x) =
2
p

Z
x
0
e
t
2
dt
(60)
Z
xe
x
dx= (x1)e
x
(61)
Z
xe
ax
dx=

x
a

1
a
2

e
ax
(62)
Z
x
2
e
x
dx=

x
2
2x+ 2

e
x
(63)
Z
x
2
e
ax
dx=

x
2
a

2x
a
2
+
2
a
3

e
ax
(64)
Z
x
3
e
x
dx=

x
3
3x
2
+ 6x6

e
x
(65)
Z
x
n
e
ax
dx=
x
n
e
ax
a

n
a
Z
x
n1
e
ax
dx
(66)
Z
x
n
e
ax
dx=
(1)
n
a
n+1
[1 +n;ax];where (a; x) =
Z
1
x
t
a1
e
t
dt
(67)
Z
e
ax
2
dx=
i
p

2
p
a
erf

ix
p
a

7

8. (68)
Z
e
ax
2
dx=
p

2
p
a
erf

x
p
a

(69)
Z
xe
ax
2
dx=
1
2a
e
ax
2
(70)
Z
x
2
e
ax
2
dx=
1
4
r

a
3
erf(x
p
a)
x
2a
e
ax
2
Integrals with Trigonometric Functions
(71)
Z
sinax dx=
1
a
cosax
(72)
Z
sin
2
ax dx=
x
2

sin 2ax
4a
(73)
Z
sin
3
ax dx=
3 cosax
4a
+
cos 3ax
12a
(74)
Z
sin
n
ax dx=
1
a
cosax2F1

1
2
;
1n
2
;
3
2
;cos
2
ax

(75)
Z
cosax dx=
1
a
sinax
(76)
Z
cos
2
ax dx=
x
2
+
sin 2ax
4a
(77)
Z
cos
3
axdx=
3 sinax
4a
+
sin 3ax
12a
8

9. (78)
Z
cos
p
axdx=
1
a(1 +p)
cos
1+p
ax2F1

1 +p
2
;
1
2
;
3 +p
2
;cos
2
ax

(79)
Z
cosxsinx dx=
1
2
sin
2
x+c1=
1
2
cos
2
x+c2=
1
4
cos 2x+c3
(80)
Z
cosaxsinbx dx=
cos[(ab)x]
2(ab)

cos[(a+b)x]
2(a+b)
; a6=b
(81)
Z
sin
2
axcosbx dx=
sin[(2ab)x]
4(2ab)
+
sinbx
2b

sin[(2a+b)x]
4(2a+b)
(82)
Z
sin
2
xcosx dx=
1
3
sin
3
x
(83)
Z
cos
2
axsinbx dx=
cos[(2ab)x]
4(2ab)

cosbx
2b

cos[(2a+b)x]
4(2a+b)
(84)
Z
cos
2
axsinax dx=
1
3a
cos
3
ax
(85)
Z
sin
2
axcos
2
bxdx=
x
4

sin 2ax
8a

sin[2(ab)x]
16(ab)
+
sin 2bx
8b

sin[2(a+b)x]
16(a+b)
(86)
Z
sin
2
axcos
2
ax dx=
x
8

sin 4ax
32a
(87)
Z
tanax dx=
1
a
ln cosax
9

10. (88)
Z
tan
2
ax dx=x+
1
a
tanax
(89)
Z
tan
n
ax dx=
tan
n+1
ax
a(1 +n)
2F1

n+ 1
2
;1;
n+ 3
2
;tan
2
ax

(90)
Z
tan
3
axdx=
1
a
ln cosax+
1
2a
sec
2
ax
(91)
Z
secx dx= lnjsecx+ tanxj= 2 tanh
1

tan
x
2

(92)
Z
sec
2
ax dx=
1
a
tanax
(93)
Z
sec
3
x dx=
1
2
secxtanx+
1
2
lnjsecx+ tanxj
(94)
Z
secxtanx dx= secx
(95)
Z
sec
2
xtanx dx=
1
2
sec
2
x
(96)
Z
sec
n
xtanx dx=
1
n
sec
n
x; n6= 0
(97)
Z
cscx dx= ln


tan
x
2


= lnjcscxcotxj+C
10

11. (98)
Z
csc
2
ax dx=
1
a
cotax
(99)
Z
csc
3
x dx=
1
2
cotxcscx+
1
2
lnjcscxcotxj
(100)
Z
csc
n
xcotx dx=
1
n
csc
n
x; n6= 0
(101)
Z
secxcscx dx= lnjtanxj
Products of Trigonometric Functions and Mono-
mials
(102)
Z
xcosx dx= cosx+xsinx
(103)
Z
xcosax dx=
1
a
2
cosax+
x
a
sinax
(104)
Z
x
2
cosx dx= 2xcosx+

x
2
2

sinx
(105)
Z
x
2
cosax dx=
2xcosax
a
2
+
a
2
x
2
2
a
3
sinax
(106)
Z
x
n
cosxdx=
1
2
(i)
n+1
[(n+ 1;ix) + (1)
n
(n+ 1; ix)]
11

12. (107)
Z
x
n
cosax dx=
1
2
(ia)
1n
[(1)
n
(n+ 1;iax)(n+ 1; ixa)]
(108)
Z
xsinx dx=xcosx+ sinx
(109)
Z
xsinax dx=
xcosax
a
+
sinax
a
2
(110)
Z
x
2
sinx dx=

2x
2

cosx+ 2xsinx
(111)
Z
x
2
sinax dx=
2a
2
x
2
a
3
cosax+
2xsinax
a
2
(112)
Z
x
n
sinx dx=
1
2
(i)
n
[(n+ 1;ix)(1)
n
(n+ 1;ix)]
(113)
Z
xcos
2
x dx=
x
2
4
+
1
8
cos 2x+
1
4
xsin 2x
(114)
Z
xsin
2
x dx=
x
2
4

1
8
cos 2x
1
4
xsin 2x
(115)
Z
xtan
2
x dx=
x
2
2
+ ln cosx+xtanx
(116)
Z
xsec
2
x dx= ln cosx+xtanx
12

13. Products of Trigonometric Functions and Ex-
ponentials
(117)
Z
e
x
sinx dx=
1
2
e
x
(sinxcosx)
(118)
Z
e
bx
sinax dx=
1
a
2
+b
2
e
bx
(bsinaxacosax)
(119)
Z
e
x
cosx dx=
1
2
e
x
(sinx+ cosx)
(120)
Z
e
bx
cosax dx=
1
a
2
+b
2
e
bx
(asinax+bcosax)
(121)
Z
xe
x
sinx dx=
1
2
e
x
(cosxxcosx+xsinx)
(122)
Z
xe
x
cosx dx=
1
2
e
x
(xcosxsinx+xsinx)
Integrals of Hyperbolic Functions
(123)
Z
coshax dx=
1
a
sinhax
(124)
Z
e
ax
coshbx dx=
8
>
<
>
:
e
ax
a
2
b
2
[acoshbxbsinhbx]a6=b
e
2ax
4a
+
x
2
a=b
(125)
Z
sinhax dx=
1
a
coshax
13

14. (126)
Z
e
ax
sinhbx dx=
8
>
<
>
:
e
ax
a
2
b
2
[bcoshbx+asinhbx]a6=b
e
2ax
4a

x
2
a=b
(127)
Z
tanhaxdx=
1
a
ln coshax
(128)
Z
e
ax
tanhbx dx=
8
>
>
>
>
>
<
>
>
>
>
>
:
e
(a+2b)x
(a+ 2b)
2F1
h
1 +
a
2b
;1;2 +
a
2b
;e
2bx
i

1
a
e
ax
2F1
h
1;
a
2b
;1 +
a
2b
;e
2bx
i
a6=b
e
ax
2 tan
1
[e
ax
]
a
a=b
(129)
Z
cosaxcoshbx dx=
1
a
2
+b
2
[asinaxcoshbx+bcosaxsinhbx]
(130)
Z
cosaxsinhbx dx=
1
a
2
+b
2
[bcosaxcoshbx+asinaxsinhbx]
(131)
Z
sinaxcoshbx dx=
1
a
2
+b
2
[acosaxcoshbx+bsinaxsinhbx]
(132)
Z
sinaxsinhbx dx=
1
a
2
+b
2
[bcoshbxsinaxacosaxsinhbx]
(133)
Z
sinhaxcoshaxdx=
1
4a
[2ax+ sinh 2ax]
(134)
Z
sinhaxcoshbx dx=
1
b
2
a
2
[bcoshbxsinhaxacoshaxsinhbx]
c2014. Fromhttp://integral-table.com, last revised June 14, 2014. This mate-
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