The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.
∫
r
d
x
=
1
2
(
x
r
+
a
2
ln
(
x
+
r
)
)
∫
r
3
d
x
=
1
4
x
r
3
+
3
8
a
2
x
r
+
3
8
a
4
ln
(
x
+
r
)
∫
r
5
d
x
=
1
6
x
r
5
+
5
24
a
2
x
r
3
+
5
16
a
4
x
r
+
5
16
a
6
ln
(
x
+
r
)
∫
x
r
d
x
=
r
3
3
∫
x
r
3
d
x
=
r
5
5
∫
x
r
2
n
+
1
d
x
=
r
2
n
+
3
2
n
+
3
∫
x
2
r
d
x
=
x
r
3
4
−
a
2
x
r
8
−
a
4
8
ln
(
x
+
r
)
∫
x
2
r
3
d
x
=
x
r
5
6
−
a
2
x
r
3
24
−
a
4
x
r
16
−
a
6
16
ln
(
x
+
r
)
∫
x
3
r
d
x
=
r
5
5
−
a
2
r
3
3
∫
x
3
r
3
d
x
=
r
7
7
−
a
2
r
5
5
∫
x
3
r
2
n
+
1
d
x
=
r
2
n
+
5
2
n
+
5
−
a
2
r
2
n
+
3
2
n
+
3
∫
x
4
r
d
x
=
x
3
r
3
6
−
a
2
x
r
3
8
+
a
4
x
r
16
+
a
6
16
ln
(
x
+
r
)
∫
x
4
r
3
d
x
=
x
3
r
5
8
−
a
2
x
r
5
16
+
a
4
x
r
3
64
+
3
a
6
x
r
128
+
3
a
8
128
ln
(
x
+
r
)
∫
x
5
r
d
x
=
r
7
7
−
2
a
2
r
5
5
+
a
4
r
3
3
∫
x
5
r
3
d
x
=
r
9
9
−
2
a
2
r
7
7
+
a
4
r
5
5
∫
x
5
r
2
n
+
1
d
x
=
r
2
n
+
7
2
n
+
7
−
2
a
2
r
2
n
+
5
2
n
+
5
+
a
4
r
2
n
+
3
2
n
+
3
∫
r
d
x
x
=
r
−
a
ln
|
a
+
r
x
|
=
r
−
a
arsinh
a
x
∫
r
3
d
x
x
=
r
3
3
+
a
2
r
−
a
3
ln
|
a
+
r
x
|
∫
r
5
d
x
x
=
r
5
5
+
a
2
r
3
3
+
a
4
r
−
a
5
ln
|
a
+
r
x
|
∫
r
7
d
x
x
=
r
7
7
+
a
2
r
5
5
+
a
4
r
3
3
+
a
6
r
−
a
7
ln
|
a
+
r
x
|
∫
d
x
r
=
arsinh
x
a
=
ln
(
x
+
r
a
)
∫
d
x
r
3
=
x
a
2
r
∫
x
d
x
r
=
r
∫
x
d
x
r
3
=
−
1
r
∫
x
2
d
x
r
=
x
2
r
−
a
2
2
arsinh
x
a
=
x
2
r
−
a
2
2
ln
(
x
+
r
a
)
∫
d
x
x
r
=
−
1
a
arsinh
a
x
=
−
1
a
ln
|
a
+
r
x
|
Assume x2 > a2 (for x2 < a2, see next section):
∫
s
d
x
=
1
2
(
x
s
−
a
2
ln
(
x
+
s
)
)
∫
x
s
d
x
=
1
3
s
3
∫
s
d
x
x
=
s
−
a
arccos
|
a
x
|
∫
d
x
s
=
ln
|
x
+
s
a
|
Here
ln
|
x
+
s
a
|
=
s
g
n
(
x
)
arcosh
|
x
a
|
=
1
2
ln
(
x
+
s
x
−
s
)
, where the positive value of
arcosh
|
x
a
|
is to be taken.
∫
x
d
x
s
=
s
∫
x
d
x
s
3
=
−
1
s
∫
x
d
x
s
5
=
−
1
3
s
3
∫
x
d
x
s
7
=
−
1
5
s
5
∫
x
d
x
s
2
n
+
1
=
−
1
(
2
n
−
1
)
s
2
n
−
1
∫
x
2
m
d
x
s
2
n
+
1
=
−
1
2
n
−
1
x
2
m
−
1
s
2
n
−
1
+
2
m
−
1
2
n
−
1
∫
x
2
m
−
2
d
x
s
2
n
−
1
∫
x
2
d
x
s
=
x
s
2
+
a
2
2
ln
|
x
+
s
a
|
∫
x
2
d
x
s
3
=
−
x
s
+
ln
|
x
+
s
a
|
∫
x
4
d
x
s
=
x
3
s
4
+
3
8
a
2
x
s
+
3
8
a
4
ln
|
x
+
s
a
|
∫
x
4
d
x
s
3
=
x
s
2
−
a
2
x
s
+
3
2
a
2
ln
|
x
+
s
a
|
∫
x
4
d
x
s
5
=
−
x
s
−
1
3
x
3
s
3
+
ln
|
x
+
s
a
|
∫
x
2
m
d
x
s
2
n
+
1
=
(
−
1
)
n
−
m
1
a
2
(
n
−
m
)
∑
i
=
0
n
−
m
−
1
1
2
(
m
+
i
)
+
1
(
n
−
m
−
1
i
)
x
2
(
m
+
i
)
+
1
s
2
(
m
+
i
)
+
1
(
n
>
m
≥
0
)
∫
d
x
s
3
=
−
1
a
2
x
s
∫
d
x
s
5
=
1
a
4
[
x
s
−
1
3
x
3
s
3
]
∫
d
x
s
7
=
−
1
a
6
[
x
s
−
2
3
x
3
s
3
+
1
5
x
5
s
5
]
∫
d
x
s
9
=
1
a
8
[
x
s
−
3
3
x
3
s
3
+
3
5
x
5
s
5
−
1
7
x
7
s
7
]
∫
x
2
d
x
s
5
=
−
1
a
2
x
3
3
s
3
∫
x
2
d
x
s
7
=
1
a
4
[
1
3
x
3
s
3
−
1
5
x
5
s
5
]
∫
x
2
d
x
s
9
=
−
1
a
6
[
1
3
x
3
s
3
−
2
5
x
5
s
5
+
1
7
x
7
s
7
]
∫
u
d
x
=
1
2
(
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
∫
x
u
d
x
=
−
1
3
u
3
(
|
x
|
≤
|
a
|
)
∫
x
2
u
d
x
=
−
x
4
u
3
+
a
2
8
(
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
∫
u
d
x
x
=
u
−
a
ln
|
a
+
u
x
|
(
|
x
|
≤
|
a
|
)
∫
d
x
u
=
arcsin
x
a
(
|
x
|
≤
|
a
|
)
∫
x
2
d
x
u
=
1
2
(
−
x
u
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
∫
u
d
x
=
1
2
(
x
u
−
sgn
x
arcosh
|
x
a
|
)
(for
|
x
|
≥
|
a
|
)
∫
x
u
d
x
=
−
u
(
|
x
|
≤
|
a
|
)
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.
∫
d
x
R
=
1
a
ln
|
2
a
R
+
2
a
x
+
b
|
(for
a
>
0
)
∫
d
x
R
=
1
a
arsinh
2
a
x
+
b
4
a
c
−
b
2
(for
a
>
0
,
4
a
c
−
b
2
>
0
)
∫
d
x
R
=
1
a
ln
|
2
a
x
+
b
|
(for
a
>
0
,
4
a
c
−
b
2
=
0
)
∫
d
x
R
=
−
1
−
a
arcsin
2
a
x
+
b
b
2
−
4
a
c
(for
a
<
0
,
4
a
c
−
b
2
<
0
,
|
2
a
x
+
b
|
<
b
2
−
4
a
c
)
∫
d
x
R
3
=
4
a
x
+
2
b
(
4
a
c
−
b
2
)
R
∫
d
x
R
5
=
4
a
x
+
2
b
3
(
4
a
c
−
b
2
)
R
(
1
R
2
+
8
a
4
a
c
−
b
2
)
∫
d
x
R
2
n
+
1
=
2
(
2
n
−
1
)
(
4
a
c
−
b
2
)
(
2
a
x
+
b
R
2
n
−
1
+
4
a
(
n
−
1
)
∫
d
x
R
2
n
−
1
)
∫
x
R
d
x
=
R
a
−
b
2
a
∫
d
x
R
∫
x
R
3
d
x
=
−
2
b
x
+
4
c
(
4
a
c
−
b
2
)
R
∫
x
R
2
n
+
1
d
x
=
−
1
(
2
n
−
1
)
a
R
2
n
−
1
−
b
2
a
∫
d
x
R
2
n
+
1
∫
d
x
x
R
=
−
1
c
ln
|
2
c
R
+
b
x
+
2
c
x
|
,
c
>
0
∫
d
x
x
R
=
−
1
c
arsinh
(
b
x
+
2
c
|
x
|
4
a
c
−
b
2
)
,
c
<
0
∫
d
x
x
R
=
1
−
c
arcsin
(
b
x
+
2
c
|
x
|
b
2
−
4
a
c
)
,
c
<
0
,
b
2
−
4
a
c
>
0
∫
d
x
x
R
=
−
2
b
x
(
a
x
2
+
b
x
)
,
c
=
0
∫
x
2
R
d
x
=
2
a
x
−
3
b
4
a
2
R
+
3
b
2
−
4
a
c
8
a
2
∫
d
x
R
∫
d
x
x
2
R
=
−
R
c
x
−
b
2
c
∫
d
x
x
R
∫
R
d
x
=
2
a
x
+
b
4
a
R
+
4
a
c
−
b
2
8
a
∫
d
x
R
∫
x
R
d
x
=
R
3
3
a
−
b
(
2
a
x
+
b
)
8
a
2
R
−
b
(
4
a
c
−
b
2
)
16
a
2
∫
d
x
R
∫
x
2
R
d
x
=
6
a
x
−
5
b
24
a
2
R
3
+
5
b
2
−
4
a
c
16
a
2
∫
R
d
x
∫
R
x
d
x
=
R
+
b
2
∫
d
x
R
+
c
∫
d
x
x
R
∫
R
x
2
d
x
=
−
R
x
+
a
∫
d
x
R
+
b
2
∫
d
x
x
R
∫
x
2
d
x
R
3
=
(
2
b
2
−
4
a
c
)
x
+
2
b
c
a
(
4
a
c
−
b
2
)
R
+
1
a
∫
d
x
R
∫
S
d
x
=
2
S
3
3
a
∫
d
x
S
=
2
S
a
∫
d
x
x
S
=
{
−
2
b
a
r
c
o
t
h
(
S
b
)
(for
b
>
0
,
a
x
>
0
)
−
2
b
a
r
t
a
n
h
(
S
b
)
(for
b
>
0
,
a
x
<
0
)
2
−
b
arctan
(
S
−
b
)
(for
b
<
0
)
∫
S
x
d
x
=
{
2
(
S
−
b
a
r
c
o
t
h
(
S
b
)
)
(for
b
>
0
,
a
x
>
0
)
2
(
S
−
b
a
r
t
a
n
h
(
S
b
)
)
(for
b
>
0
,
a
x
<
0
)
2
(
S
−
−
b
arctan
(
S
−
b
)
)
(for
b
<
0
)
∫
x
n
S
d
x
=
2
a
(
2
n
+
1
)
(
x
n
S
−
b
n
∫
x
n
−
1
S
d
x
)
∫
x
n
S
d
x
=
2
a
(
2
n
+
3
)
(
x
n
S
3
−
n
b
∫
x
n
−
1
S
d
x
)
∫
1
x
n
S
d
x
=
−
1
b
(
n
−
1
)
(
S
x
n
−
1
+
(
n
−
3
2
)
a
∫
d
x
x
n
−
1
S
)