The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.
Generally, if the function
sin
(
x
)
is any trigonometric function, and
cos
(
x
)
is its derivative,
∫
a
cos
n
x
d
x
=
a
n
sin
n
x
+
C
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
Integrands involving only sine
∫
sin
a
x
d
x
=
−
1
a
cos
a
x
+
C
∫
sin
2
a
x
d
x
=
x
2
−
1
4
a
sin
2
a
x
+
C
=
x
2
−
1
2
a
sin
a
x
cos
a
x
+
C
∫
sin
3
a
x
d
x
=
cos
3
a
x
12
a
−
3
cos
a
x
4
a
+
C
∫
x
sin
2
a
x
d
x
=
x
2
4
−
x
4
a
sin
2
a
x
−
1
8
a
2
cos
2
a
x
+
C
∫
x
2
sin
2
a
x
d
x
=
x
3
6
−
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
−
x
4
a
2
cos
2
a
x
+
C
∫
sin
b
1
x
sin
b
2
x
d
x
=
sin
(
(
b
2
−
b
1
)
x
)
2
(
b
2
−
b
1
)
−
sin
(
(
b
1
+
b
2
)
x
)
2
(
b
1
+
b
2
)
+
C
(for
|
b
1
|
≠
|
b
2
|
)
∫
sin
n
a
x
d
x
=
−
sin
n
−
1
a
x
cos
a
x
n
a
+
n
−
1
n
∫
sin
n
−
2
a
x
d
x
(for
n
>
0
)
∫
d
x
sin
a
x
=
−
1
a
ln
|
csc
a
x
+
cot
a
x
|
+
C
∫
d
x
sin
n
a
x
=
cos
a
x
a
(
1
−
n
)
sin
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
sin
n
−
2
a
x
(for
n
>
1
)
∫
x
sin
a
x
d
x
=
sin
a
x
a
2
−
x
cos
a
x
a
+
C
∫
x
n
sin
a
x
d
x
=
−
x
n
a
cos
a
x
+
n
a
∫
x
n
−
1
cos
a
x
d
x
=
∑
k
=
0
2
k
≤
n
(
−
1
)
k
+
1
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
cos
a
x
+
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
1
−
2
k
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
sin
a
x
(for
n
>
0
)
∫
sin
a
x
x
d
x
=
∑
n
=
0
∞
(
−
1
)
n
(
a
x
)
2
n
+
1
(
2
n
+
1
)
⋅
(
2
n
+
1
)
!
+
C
∫
sin
a
x
x
n
d
x
=
−
sin
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
∫
cos
a
x
x
n
−
1
d
x
∫
d
x
1
±
sin
a
x
=
1
a
tan
(
a
x
2
∓
π
4
)
+
C
∫
x
d
x
1
+
sin
a
x
=
x
a
tan
(
a
x
2
−
π
4
)
+
2
a
2
ln
|
cos
(
a
x
2
−
π
4
)
|
+
C
∫
x
d
x
1
−
sin
a
x
=
x
a
cot
(
π
4
−
a
x
2
)
+
2
a
2
ln
|
sin
(
π
4
−
a
x
2
)
|
+
C
∫
sin
a
x
d
x
1
±
sin
a
x
=
±
x
+
1
a
tan
(
π
4
∓
a
x
2
)
+
C
Integrands involving only cosine
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
C
∫
cos
2
a
x
d
x
=
x
2
+
1
4
a
sin
2
a
x
+
C
=
x
2
+
1
2
a
sin
a
x
cos
a
x
+
C
∫
cos
n
a
x
d
x
=
cos
n
−
1
a
x
sin
a
x
n
a
+
n
−
1
n
∫
cos
n
−
2
a
x
d
x
(for
n
>
0
)
∫
x
cos
a
x
d
x
=
cos
a
x
a
2
+
x
sin
a
x
a
+
C
∫
x
2
cos
2
a
x
d
x
=
x
3
6
+
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
+
x
4
a
2
cos
2
a
x
+
C
∫
x
n
cos
a
x
d
x
=
x
n
sin
a
x
a
−
n
a
∫
x
n
−
1
sin
a
x
d
x
=
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
2
k
−
1
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
cos
a
x
+
∑
k
=
0
2
k
≤
n
(
−
1
)
k
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
sin
a
x
∫
cos
a
x
x
d
x
=
ln
|
a
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
a
x
)
2
k
2
k
⋅
(
2
k
)
!
+
C
∫
cos
a
x
x
n
d
x
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
∫
sin
a
x
x
n
−
1
d
x
(for
n
≠
1
)
∫
d
x
cos
a
x
=
1
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
∫
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
>
1
)
∫
d
x
1
+
cos
a
x
=
1
a
tan
a
x
2
+
C
∫
d
x
1
−
cos
a
x
=
−
1
a
cot
a
x
2
+
C
∫
x
d
x
1
+
cos
a
x
=
x
a
tan
a
x
2
+
2
a
2
ln
|
cos
a
x
2
|
+
C
∫
x
d
x
1
−
cos
a
x
=
−
x
a
cot
a
x
2
+
2
a
2
ln
|
sin
a
x
2
|
+
C
∫
cos
a
x
d
x
1
+
cos
a
x
=
x
−
1
a
tan
a
x
2
+
C
∫
cos
a
x
d
x
1
−
cos
a
x
=
−
x
−
1
a
cot
a
x
2
+
C
∫
cos
a
1
x
cos
a
2
x
d
x
=
sin
(
(
a
2
−
a
1
)
x
)
2
(
a
2
−
a
1
)
+
sin
(
(
a
2
+
a
1
)
x
)
2
(
a
2
+
a
1
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
Integrands involving only tangent
∫
tan
a
x
d
x
=
−
1
a
ln
|
cos
a
x
|
+
C
=
1
a
ln
|
sec
a
x
|
+
C
∫
tan
2
x
d
x
=
tan
x
−
x
+
C
∫
tan
n
a
x
d
x
=
1
a
(
n
−
1
)
tan
n
−
1
a
x
−
∫
tan
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
d
x
q
tan
a
x
+
p
=
1
p
2
+
q
2
(
p
x
+
q
a
ln
|
q
sin
a
x
+
p
cos
a
x
|
)
+
C
(for
p
2
+
q
2
≠
0
)
∫
d
x
tan
a
x
+
1
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
∫
d
x
tan
a
x
−
1
=
−
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
∫
tan
a
x
d
x
tan
a
x
+
1
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
∫
tan
a
x
d
x
tan
a
x
−
1
=
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
Integrands involving only secant
See Integral of the secant function.
∫
sec
a
x
d
x
=
1
a
ln
|
sec
a
x
+
tan
a
x
|
+
C
∫
sec
2
x
d
x
=
tan
x
+
C
∫
sec
3
x
d
x
=
1
2
sec
x
tan
x
+
1
2
ln
|
sec
x
+
tan
x
|
+
C
.
∫
sec
n
a
x
d
x
=
sec
n
−
2
a
x
tan
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
sec
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
d
x
sec
x
+
1
=
x
−
tan
x
2
+
C
∫
d
x
sec
x
−
1
=
−
x
−
cot
x
2
+
C
Integrands involving only cosecant
∫
csc
a
x
d
x
=
−
1
a
ln
|
csc
a
x
+
cot
a
x
|
+
C
∫
csc
2
x
d
x
=
−
cot
x
+
C
∫
csc
3
x
d
x
=
−
1
2
csc
x
cot
x
−
1
2
ln
|
csc
x
+
cot
x
|
+
C
.
∫
csc
n
a
x
d
x
=
−
csc
n
−
2
a
x
cot
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
csc
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
d
x
csc
x
+
1
=
x
−
2
cot
x
2
+
1
+
C
∫
d
x
csc
x
−
1
=
−
x
+
2
cot
x
2
−
1
+
C
Integrands involving only cotangent
∫
cot
a
x
d
x
=
1
a
ln
|
sin
a
x
|
+
C
∫
cot
n
a
x
d
x
=
−
1
a
(
n
−
1
)
cot
n
−
1
a
x
−
∫
cot
n
−
2
a
x
d
x
(for
n
≠
1
)
∫
d
x
1
+
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
+
1
∫
d
x
1
−
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
−
1
Integrands involving both sine and cosine
∫
d
x
cos
a
x
±
sin
a
x
=
1
a
2
ln
|
tan
(
a
x
2
±
π
8
)
|
+
C
∫
d
x
(
cos
a
x
±
sin
a
x
)
2
=
1
2
a
tan
(
a
x
∓
π
4
)
+
C
∫
d
x
(
cos
x
+
sin
x
)
n
=
1
n
−
1
(
sin
x
−
cos
x
(
cos
x
+
sin
x
)
n
−
1
−
2
(
n
−
2
)
∫
d
x
(
cos
x
+
sin
x
)
n
−
2
)
∫
cos
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
∫
cos
a
x
d
x
cos
a
x
−
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
∫
sin
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
∫
sin
a
x
d
x
cos
a
x
−
sin
a
x
=
−
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
∫
cos
a
x
d
x
sin
a
x
(
1
+
cos
a
x
)
=
−
1
4
a
tan
2
a
x
2
+
1
2
a
ln
|
tan
a
x
2
|
+
C
∫
cos
a
x
d
x
sin
a
x
(
1
−
cos
a
x
)
=
−
1
4
a
cot
2
a
x
2
−
1
2
a
ln
|
tan
a
x
2
|
+
C
∫
sin
a
x
d
x
cos
a
x
(
1
+
sin
a
x
)
=
1
4
a
cot
2
(
a
x
2
+
π
4
)
+
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
∫
sin
a
x
d
x
cos
a
x
(
1
−
sin
a
x
)
=
1
4
a
tan
2
(
a
x
2
+
π
4
)
−
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
∫
sin
a
x
cos
a
x
d
x
=
1
2
a
sin
2
a
x
+
C
∫
sin
a
1
x
cos
a
2
x
d
x
=
−
cos
(
(
a
1
−
a
2
)
x
)
2
(
a
1
−
a
2
)
−
cos
(
(
a
1
+
a
2
)
x
)
2
(
a
1
+
a
2
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
∫
sin
n
a
x
cos
a
x
d
x
=
1
a
(
n
+
1
)
sin
n
+
1
a
x
+
C
(for
n
≠
−
1
)
∫
sin
a
x
cos
n
a
x
d
x
=
−
1
a
(
n
+
1
)
cos
n
+
1
a
x
+
C
(for
n
≠
−
1
)
∫
sin
n
a
x
cos
m
a
x
d
x
=
−
sin
n
−
1
a
x
cos
m
+
1
a
x
a
(
n
+
m
)
+
n
−
1
n
+
m
∫
sin
n
−
2
a
x
cos
m
a
x
d
x
(for
m
,
n
>
0
)
also:
∫
sin
n
a
x
cos
m
a
x
d
x
=
sin
n
+
1
a
x
cos
m
−
1
a
x
a
(
n
+
m
)
+
m
−
1
n
+
m
∫
sin
n
a
x
cos
m
−
2
a
x
d
x
(for
m
,
n
>
0
)
∫
d
x
sin
a
x
cos
a
x
=
1
a
ln
|
tan
a
x
|
+
C
∫
d
x
sin
a
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
∫
d
x
sin
a
x
cos
n
−
2
a
x
(for
n
≠
1
)
∫
d
x
sin
n
a
x
cos
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
∫
d
x
sin
n
−
2
a
x
cos
a
x
(for
n
≠
1
)
∫
sin
a
x
d
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
C
(for
n
≠
1
)
∫
sin
2
a
x
d
x
cos
a
x
=
−
1
a
sin
a
x
+
1
a
ln
|
tan
(
π
4
+
a
x
2
)
|
+
C
∫
sin
2
a
x
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
1
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
≠
1
)
∫
sin
n
a
x
d
x
cos
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
1
)
+
∫
sin
n
−
2
a
x
d
x
cos
a
x
(for
n
≠
1
)
∫
sin
n
a
x
d
x
cos
m
a
x
=
sin
n
+
1
a
x
a
(
m
−
1
)
cos
m
−
1
a
x
−
n
−
m
+
2
m
−
1
∫
sin
n
a
x
d
x
cos
m
−
2
a
x
(for
m
≠
1
)
also:
∫
sin
n
a
x
d
x
cos
m
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
m
)
cos
m
−
1
a
x
+
n
−
1
n
−
m
∫
sin
n
−
2
a
x
d
x
cos
m
a
x
(for
m
≠
n
)
also:
∫
sin
n
a
x
d
x
cos
m
a
x
=
sin
n
−
1
a
x
a
(
m
−
1
)
cos
m
−
1
a
x
−
n
−
1
m
−
1
∫
sin
n
−
2
a
x
d
x
cos
m
−
2
a
x
(for
m
≠
1
)
∫
cos
a
x
d
x
sin
n
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
C
(for
n
≠
1
)
∫
cos
2
a
x
d
x
sin
a
x
=
1
a
(
cos
a
x
+
ln
|
tan
a
x
2
|
)
+
C
∫
cos
2
a
x
d
x
sin
n
a
x
=
−
1
n
−
1
(
cos
a
x
a
sin
n
−
1
a
x
+
∫
d
x
sin
n
−
2
a
x
)
(for
n
≠
1
)
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
+
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
m
+
2
m
−
1
∫
cos
n
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
cos
n
−
1
a
x
a
(
n
−
m
)
sin
m
−
1
a
x
+
n
−
1
n
−
m
∫
cos
n
−
2
a
x
d
x
sin
m
a
x
(for
m
≠
n
)
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
−
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
1
m
−
1
∫
cos
n
−
2
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
Integrands involving both sine and tangent
∫
sin
a
x
tan
a
x
d
x
=
1
a
(
ln
|
sec
a
x
+
tan
a
x
|
−
sin
a
x
)
+
C
∫
tan
n
a
x
d
x
sin
2
a
x
=
1
a
(
n
−
1
)
tan
n
−
1
(
a
x
)
+
C
(for
n
≠
1
)
Integrand involving both cosine and tangent
∫
tan
n
a
x
d
x
cos
2
a
x
=
1
a
(
n
+
1
)
tan
n
+
1
a
x
+
C
(for
n
≠
−
1
)
Integrand involving both sine and cotangent
∫
cot
n
a
x
d
x
sin
2
a
x
=
−
1
a
(
n
+
1
)
cot
n
+
1
a
x
+
C
(for
n
≠
−
1
)
Integrand involving both cosine and cotangent
∫
cot
n
a
x
d
x
cos
2
a
x
=
1
a
(
1
−
n
)
tan
1
−
n
a
x
+
C
(for
n
≠
1
)
Integrand involving both secant and tangent
∫
sec
x
tan
x
d
x
=
sec
x
+
C
Integrand involving with cosecant and cotangent
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
∫
0
π
2
sin
n
x
d
x
=
∫
0
π
2
cos
n
x
d
x
=
{
n
−
1
n
⋅
n
−
3
n
−
2
⋯
3
4
⋅
1
2
⋅
π
2
,
if
n
is even
n
−
1
n
⋅
n
−
3
n
−
2
⋯
4
5
⋅
2
3
,
if
n
is odd and more than 1
1
,
if
n
=
1
∫
−
c
c
sin
x
d
x
=
0
∫
−
c
c
cos
x
d
x
=
2
∫
0
c
cos
x
d
x
=
2
∫
−
c
0
cos
x
d
x
=
2
sin
c
∫
−
c
c
tan
x
d
x
=
0
∫
−
a
2
a
2
x
2
cos
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
)
24
n
2
π
2
(for
n
=
1
,
3
,
5...
)
∫
−
a
2
a
2
x
2
sin
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
(
−
1
)
n
)
24
n
2
π
2
=
a
3
24
(
1
−
6
(
−
1
)
n
n
2
π
2
)
(for
n
=
1
,
2
,
3
,
.
.
.
)
∫
0
2
π
sin
2
m
+
1
x
cos
2
n
+
1
x
d
x
=
0
{
n
,
m
}
∈
Z
List of integrals of trigonometric functions Wikipedia (Text) CC BY-SA
