Integral table

INtegral

1. Table of Integrals

Basic Forms
Z
x
n
dx=
1
n+ 1
x
n+1
+c (1)
Z
1
x
dx= lnx+c (2)
Z
udv=uv
Z
vdu (3)
Z
1
ax+b
dx=
1
a
lnjax+bj+c (4)
Integrals of Rational Functions
Z
1
(x+a)
2
dx=
1
x+a
+c (5)
Z
(x+a)
n
dx=
(x+a)
n+1
n+ 1
+c; n6=1 (6)
Z
x(x+a)
n
dx=
(x+a)
n+1
((n+ 1)xa)
(n+ 1)(n+ 2)
+c(7)
Z
1
1 +x
2
dx= tan
1
x+c (8)
Z
1
a
2
+x
2
dx=
1
a
tan
1
x
a
+c (9)
Z
x
a
2
+x
2
dx=
1
2
lnja
2
+x
2
j+c (10)
Z
x
2
a
2
+x
2
dx=xatan
1
x
a
+c (11)
Z
x
3
a
2
+x
2
dx=
1
2
x
2

1
2
a
2
lnja
2
+x
2
j+c (12)
Z
1
ax
2
+bx+c
dx=
2
p
4acb
2
tan
1
2ax+b
p
4acb
2
+C(13)
Z
1
(x+a)(x+b)
dx=
1
ba
ln
a+x
b+x
; a6=b(14)
Z
x
(x+a)
2
dx=
a
a+x
+ lnja+xj+C (15)
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
+C(16)
Integrals with Roots
Z
p
xadx=
2
3
(xa)
3=2
+C (17)
Z
1
p
xa
dx= 2
p
xa+C (18)
Z
1
p
ax
dx= 2
p
ax+C (19)
Z
x
p
xadx=
2
3
a(xa)
3=2
+
2
5
(xa)
5=2
+C (20)
Z
p
ax+bdx=

2b
3a
+
2x
3

p
ax+b+C (21)
Z
(ax+b)
3=2
dx=
2
5a
(ax+b)
5=2
+C (22)
Z
x
p
xa
dx=
2
3
(x2a)
p
xa+C (23)
Zr
x
ax
dx=
p
x(ax)
atan
1
p
x(ax)
xa
+C (24)
Zr
x
a+x
dx=
p
x(a+x)
aln
p
x+
p
x+a

+C (25)
Z
x
p
ax+bdx=
2
15a
2
(2b
2
+abx+ 3a
2
x
2
)
p
ax+b+C (26)
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
+C(27)
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)


+C(28)

c2007. From http://integral-table.com, last revised December 6, 2007. This material is provided as is without warranty or representation
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terms of the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. See the web site for details. The formula numbers
on this document may be dierent from the formula numbers on the web page.
1

2. 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


+C (29)
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
+C (30)
Z
x
p
x
2
a
2
dx=
1
3

x
2
a
2

3=2
+C (31)
Z
1
p
x
2
a
2
dx= ln


x+
p
x
2
a
2


+C (32)
Z
1
p
a
2
x
2
dx= sin
1
x
a
+C (33)
Z
x
p
x
2
a
2
dx=
p
x
2
a
2
+C (34)
Z
x
p
a
2
x
2
dx=
p
a
2
x
2
+C (35)
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


+C (36)
Z
p
ax
2
+bx+cdx=
b+ 2ax
4a
p
ax
2
+bx+c
+
4acb
2
8a
3=2
ln


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


+C(37)
Z
x
p
ax
2
+bx+c=
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+x




(38)
Z
1
p
ax
2
+bx+c
dx=
1
p
a
ln


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


+C (39)
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)


+C (40)
Integrals with Logarithms
Z
lnaxdx=xlnaxx+C (41)
Z
lnax
x
dx=
1
2
(lnax)
2
+C (42)
Z
ln(ax+b)dx=

x+
b
a

ln(ax+b)x+C; a6= 0 (43)
Z
ln

a
2
x
2
b
2

dx=xln

a
2
x
2
b
2

+
2b
a
tan
1
ax
b
2x+C (44)
Z
ln

a
2
b
2
x
2

dx=xln

arb
2
x
2

+
2a
b
tan
1
bx
a
2x+C (45)
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

+C (46)
Z
xln(ax+b)dx=
bx
2a

1
4
x
2
+
1
2

x
2

b
2
a
2

ln(ax+b) +C (47)
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

+C(48)
Integrals with Exponentials
Z
e
ax
dx=
1
a
e
ax
+C (49)
Z
p
xe
ax
dx=
1
a
p
xe
ax
+
i
p

2a
3=2
erf

i
p
ax

+C;
where erf(x) =
2
p

Z
x
0
e
t
2
dtet (50)
Z
xe
x
dx= (x1)e
x
+C (51)
Z
xe
ax
dx=

x
a

1
a
2

e
ax
+C (52)
Z
x
2
e
x
dx=

x
2
2x+ 2

e
x
+C (53)
2

3. Z
x
2
e
ax
dx=

x
2
a

2x
a
2
+
2
a
3

e
ax
+C (54)
Z
x
3
e
x
dx=

x
3
3x
2
+ 6x6

e
x
+C (55)
Z
x
n
e
ax
dx= (1)
n
1
a
[1 +n;ax];
where (a; x) =
Z
1
x
t
a1
e
t
dt (56)
Z
e
ax
2
dx=
i
p

2
p
a
erf

ix
p
a

(57)
Integrals with Trigonometric Functions
Z
sinaxdx=
1
a
cosax+C (58)
Z
sin
2
axdx=
x
2

sin 2ax
4a
+C (59)
Z
sin
n
axdx=

1
a
cosax2F1

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

+C (60)
Z
sin
3
axdx=
3 cosax
4a
+
cos 3ax
12a
+C (61)
Z
cosaxdx=
1
a
sinax+C (62)
Z
cos
2
axdx=
x
2
+
sin 2ax
4a
+C (63)
Z
cos
p
axdx=
1
a(1 +p)
cos
1+p
ax
2F1

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

+C (64)
Z
cos
3
axdx=
3 sinax
4a
+
sin 3ax
12a
+C (65)
Z
cosaxsinbxdx=
cos[(ab)x]
2(ab)

cos[(a+b)x]
2(a+b)
+C; a6=b (66)
Z
sin
2
axcosbxdx=
sin[(2ab)x]
4(2ab)
+
sinbx
2b

sin[(2a+b)x]
4(2a+b)
+C (67)
Z
sin
2
xcosxdx=
1
3
sin
3
x+C (68)
Z
cos
2
axsinbxdx=
cos[(2ab)x]
4(2ab)

cosbx
2b

cos[(2a+b)x]
4(2a+b)
+C (69)
Z
cos
2
axsinaxdx=
1
3a
cos
3
ax+C (70)
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)
+C(71)
Z
sin
2
axcos
2
axdx=
x
8

sin 4ax
32a
+C (72)
Z
tanaxdx=
1
a
ln cosax+C (73)
Z
tan
2
axdx=x+
1
a
tanax+C (74)
Z
tan
n
axdx=
tan
n+1
ax
a(1 +n)

2F1

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

+C (75)
Z
tan
3
axdx=
1
a
ln cosax+
1
2a
sec
2
ax+C (76)
Z
secxdx= lnjsecx+ tanxj+C
= 2 tanh
1

tan
x
2

+C (77)
Z
sec
2
axdx=
1
a
tanax+C (78)
Z
sec
3
xdx=
1
2
secxtanx+
1
2
lnjsecxtanxj+C(79)
Z
secxtanxdx= secx+C (80)
Z
sec
2
xtanxdx=
1
2
sec
2
x+C (81)
Z
sec
n
xtanxdx=
1
n
sec
n
x+C; n6= 0 (82)
Z
cscxdx= ln


tan
x
2


+C= lnjcscxcotxj+C(83)
3

4. Z
csc
2
axdx=
1
a
cotax+C (84)
Z
csc
3
xdx=
1
2
cotxcscx+
1
2
lnjcscxcotxj+C(85)
Z
csc
n
xcotxdx=
1
n
csc
n
x+C; n6= 0 (86)
Z
secxcscxdx= lnjtanxj+C (87)
Products of Trigonometric Functions and Monomials
Z
xcosxdx= cosx+xsinx+C (88)
Z
xcosaxdx=
1
a
2
cosax+
x
a
sinax+C (89)
Z
x
2
cosxdx= 2xcosx+

x
2
2

sinx+C (90)
Z
x
2
cosaxdx=
2xcosax
a
2
+
a
2
x
2
2
a
3
sinax+C (91)
Z
x
n
cosxdx=
1
2
(i)
n+1
[(n+ 1;ix)
+(1)
n
(n+ 1; ix)] +C (92)
Z
x
n
cosaxdx=
1
2
(ia)
1n
[(1)
n
(n+ 1;iax)
(n+ 1; ixa)] +C (93)
Z
xsinxdx=xcosx+ sinx+C (94)
Z
xsinaxdx=
xcosax
a
+
sinax
a
2
+C (95)
Z
x
2
sinxdx=

2x
2

cosx+ 2xsinx+C (96)
Z
x
2
sinaxdx=
2a
2
x
2
a
3
cosax+
2xsinax
a
3
+C (97)
Z
x
n
sinxdx=
1
2
(i)
n
[(n+ 1;ix)
(1)
n
(n+ 1;ix)] +C (98)
Products of Trigonometric Functions and
Exponentials
Z
e
x
sinxdx=
1
2
e
x
(sinxcosx) +C (99)
Z
e
bx
sinaxdx=
1
a
2
+b
2
e
bx
(bsinaxacosbx) +C(100)
Z
e
x
cosxdx=
1
2
e
x
(sinx+ cosx) +C (101)
Z
e
bx
cosaxdx=
1
a
2
+b
2
e
bx
(asinax+bcosax) +C(102)
Z
xe
x
sinxdx=
1
2
e
x
(cosxxcosx+xsinx) +C(103)
Z
xe
x
cosxdx=
1
2
e
x
(xcosxsinx+xsinx) +C(104)
Integrals of Hyperbolic Functions
Z
coshaxdx=
1
a
sinhax+C (105)
Z
e
ax
coshbxdx=
8
>
<
>
:
e
ax
a
2
b
2
[acoshbxbsinhbx] +C a6=b
e
2ax
4a
+
x
2
+C a =b
(106)
Z
sinhaxdx=
1
a
coshax+C (107)
Z
e
ax
sinhbxdx=
8
>
<
>
:
e
ax
a
2
b
2
[bcoshbx+asinhbx] +C a6=b
e
2ax
4a

x
2
+C a =b
(108)
Z
e
ax
tanhbxdx=
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
a
2b
;1;1E;e
2bx
i
+C a6=b
e
ax
2 tan
1
[e
ax
]
a
+C a =b
(109)
Z
tanhbxdx=
1
a
ln coshax+C (110)
Z
cosaxcoshbxdx=
1
a
2
+b
2
[asinaxcoshbx
+bcosaxsinhbx] +C (111)
4

5. Z
cosaxsinhbxdx=
1
a
2
+b
2
[bcosaxcoshbx+
asinaxsinhbx] +C (112)
Z
sinaxcoshbxdx=
1
a
2
+b
2
[acosaxcoshbx+
bsinaxsinhbx] +C (113)
Z
sinaxsinhbxdx=
1
a
2
+b
2
[bcoshbxsinax
acosaxsinhbx] +C (114)
Z
sinhaxcoshaxdx=
1
4a
[2ax+ sinh 2ax] +C(115)
Z
sinhaxcoshbxdx=
1
b
2
a
2
[bcoshbxsinhax
acoshaxsinhbx] +C (116)
5