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 .[1]
Generally, if the function
sin
x
{\displaystyle \sin x}
is any trigonometric function, and
cos
x
{\displaystyle \cos x}
is its derivative,
∫
a
cos
n
x
d
x
=
a
n
sin
n
x
+
C
{\displaystyle \int a\cos nx\,dx={\frac {a}{n))\sin nx+C}
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
Integrands involving both sine and cosine
An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules .
∫
d
x
cos
a
x
±
sin
a
x
=
1
a
2
ln
|
tan
(
a
x
2
±
π
8
)
|
+
C
{\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax))={\frac {1}{a{\sqrt {2))))\ln \left|\tan \left({\frac {ax}{2))\pm {\frac {\pi }{8))\right)\right|+C}
∫
d
x
(
cos
a
x
±
sin
a
x
)
2
=
1
2
a
tan
(
a
x
∓
π
4
)
+
C
{\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2))}={\frac {1}{2a))\tan \left(ax\mp {\frac {\pi }{4))\right)+C}
∫
d
x
(
cos
x
+
sin
x
)
n
=
1
2
(
n
−
1
)
(
sin
x
−
cos
x
(
cos
x
+
sin
x
)
n
−
1
+
(
n
−
2
)
∫
d
x
(
cos
x
+
sin
x
)
n
−
2
)
{\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n))}={\frac {1}{2(n-1)))\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1))}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2))}\right)}
∫
cos
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\cos ax\,dx}{\cos ax+\sin ax))={\frac {x}{2))+{\frac {1}{2a))\ln \left|\sin ax+\cos ax\right|+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
{\displaystyle \int {\frac {\cos ax\,dx}{\cos ax-\sin ax))={\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax-\cos ax\right|+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
{\displaystyle \int {\frac {\sin ax\,dx}{\cos ax+\sin ax))={\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax+\cos ax\right|+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
{\displaystyle \int {\frac {\sin ax\,dx}{\cos ax-\sin ax))=-{\frac {x}{2))-{\frac {1}{2a))\ln \left|\sin ax-\cos ax\right|+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
{\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)))=-{\frac {1}{4a))\tan ^{2}{\frac {ax}{2))+{\frac {1}{2a))\ln \left|\tan {\frac {ax}{2))\right|+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
{\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)))=-{\frac {1}{4a))\cot ^{2}{\frac {ax}{2))-{\frac {1}{2a))\ln \left|\tan {\frac {ax}{2))\right|+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
{\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)))={\frac {1}{4a))\cot ^{2}\left({\frac {ax}{2))+{\frac {\pi }{4))\right)+{\frac {1}{2a))\ln \left|\tan \left({\frac {ax}{2))+{\frac {\pi }{4))\right)\right|+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
{\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)))={\frac {1}{4a))\tan ^{2}\left({\frac {ax}{2))+{\frac {\pi }{4))\right)-{\frac {1}{2a))\ln \left|\tan \left({\frac {ax}{2))+{\frac {\pi }{4))\right)\right|+C}
∫
(
sin
a
x
)
(
cos
a
x
)
d
x
=
1
2
a
sin
2
a
x
+
C
{\displaystyle \int (\sin ax)(\cos ax)\,dx={\frac {1}{2a))\sin ^{2}ax+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
|
)
{\displaystyle \int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})))-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})))+C\qquad {\mbox{(for ))|a_{1}|\neq |a_{2}|{\mbox{)))}
∫
(
sin
n
a
x
)
(
cos
a
x
)
d
x
=
1
a
(
n
+
1
)
sin
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)))\sin ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))}
∫
(
sin
a
x
)
(
cos
n
a
x
)
d
x
=
−
1
a
(
n
+
1
)
cos
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)))\cos ^{n+1}ax+C\qquad {\mbox{(for ))n\neq -1{\mbox{)))}
∫
(
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
)
=
(
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
)
{\displaystyle {\begin{aligned}\int (\sin ^{n}ax)(\cos ^{m}ax)\,dx&=-{\frac {(\sin ^{n-1}ax)(\cos ^{m+1}ax)}{a(n+m)))+{\frac {n-1}{n+m))\int (\sin ^{n-2}ax)(\cos ^{m}ax)\,dx\qquad {\mbox{(for ))m,n>0{\mbox{)))\\&={\frac {(\sin ^{n+1}ax)(\cos ^{m-1}ax)}{a(n+m)))+{\frac {m-1}{n+m))\int (\sin ^{n}ax)(\cos ^{m-2}ax)\,dx\qquad {\mbox{(for ))m,n>0{\mbox{)))\end{aligned))}
∫
d
x
(
sin
a
x
)
(
cos
a
x
)
=
1
a
ln
|
tan
a
x
|
+
C
{\displaystyle \int {\frac {dx}{(\sin ax)(\cos ax)))={\frac {1}{a))\ln \left|\tan ax\right|+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
)
{\displaystyle \int {\frac {dx}{(\sin ax)(\cos ^{n}ax)))={\frac {1}{a(n-1)\cos ^{n-1}ax))+\int {\frac {dx}{(\sin ax)(\cos ^{n-2}ax)))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
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
)
{\displaystyle \int {\frac {dx}{(\sin ^{n}ax)(\cos ax)))=-{\frac {1}{a(n-1)\sin ^{n-1}ax))+\int {\frac {dx}{(\sin ^{n-2}ax)(\cos ax)))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
sin
a
x
d
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ax\,dx}{\cos ^{n}ax))={\frac {1}{a(n-1)\cos ^{n-1}ax))+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
sin
2
a
x
d
x
cos
a
x
=
−
1
a
sin
a
x
+
1
a
ln
|
tan
(
π
4
+
a
x
2
)
|
+
C
{\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ax))=-{\frac {1}{a))\sin ax+{\frac {1}{a))\ln \left|\tan \left({\frac {\pi }{4))+{\frac {ax}{2))\right)\right|+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
)
{\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax))={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax))-{\frac {1}{n-1))\int {\frac {dx}{\cos ^{n-2}ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
sin
2
x
1
+
cos
2
x
d
x
=
2
arctangant
(
tan
x
2
)
−
x
(for x in
]
−
π
2
;
+
π
2
[
)
=
2
arctangant
(
tan
x
2
)
−
arctangant
(
tan
x
)
(this time x being any real number
)
{\displaystyle {\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x))\,dx&={\sqrt {2))\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2))}\right)-x\qquad {\mbox{(for x in))]-{\frac {\pi }{2));+{\frac {\pi }{2))[{\mbox{)))\\&={\sqrt {2))\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2))}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number )){\mbox{)))\end{aligned))}
∫
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
)
{\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ax))=-{\frac {\sin ^{n-1}ax}{a(n-1)))+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax))\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
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
)
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
)
−
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
)
{\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax))={\begin{cases}{\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax))-{\frac {n-m+2}{m-1))\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\{\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax))-{\frac {n-1}{m-1))\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m}ax))&{\mbox{(for ))m\neq n{\mbox{)))\end{cases))}
∫
cos
a
x
d
x
sin
n
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\cos ax\,dx}{\sin ^{n}ax))=-{\frac {1}{a(n-1)\sin ^{n-1}ax))+C\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
cos
2
a
x
d
x
sin
a
x
=
1
a
(
cos
a
x
+
ln
|
tan
a
x
2
|
)
+
C
{\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ax))={\frac {1}{a))\left(\cos ax+\ln \left|\tan {\frac {ax}{2))\right|\right)+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
)
{\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax))=-{\frac {1}{n-1))\left({\frac {\cos ax}{a\sin ^{n-1}ax))+\int {\frac {dx}{\sin ^{n-2}ax))\right)\qquad {\mbox{(for ))n\neq 1{\mbox{)))}
∫
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
n
≠
1
)
−
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
)
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
)
{\displaystyle \int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax))={\begin{cases}-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax))-{\frac {n-m+2}{m-1))\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m-2}ax))&{\mbox{(for ))n\neq 1{\mbox{)))\\-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax))-{\frac {n-1}{m-1))\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax))&{\mbox{(for ))m\neq 1{\mbox{)))\\{\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax))+{\frac {n-1}{n-m))\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax))&{\mbox{(for ))m\neq n{\mbox{)))\end{cases))}
Integrals in a quarter period
Using the beta function
B
(
a
,
b
)
{\displaystyle B(a,b)}
one can write
∫
0
π
2
sin
n
x
d
x
=
∫
0
π
2
cos
n
x
d
x
=
1
2
B
(
n
+
1
2
,
1
2
)
=
{
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
{\displaystyle \int _{0}^{\frac {\pi }{2))\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2))\cos ^{n}x\,dx={\frac {1}{2))B\left({\frac {n+1}{2)),{\frac {1}{2))\right)={\begin{cases}{\frac {n-1}{n))\cdot {\frac {n-3}{n-2))\cdots {\frac {3}{4))\cdot {\frac {1}{2))\cdot {\frac {\pi }{2)),&{\text{if ))n{\text{ is even))\\{\frac {n-1}{n))\cdot {\frac {n-3}{n-2))\cdots {\frac {4}{5))\cdot {\frac {2}{3)),&{\text{if ))n{\text{ is odd and more than 1))\\1,&{\text{if ))n=1\end{cases))}