Таблиця інтегралів тригонометричних функцій
Це список інтегралів (первісних функцій ) тригонометричних функцій . Для функцій що містять і показникові й тригонометричні функції, дивись Таблиця інтегралів експоненціальних функцій . Для повнішого списку інтегралів дивись Таблиця інтегралів . Дивись також тригонометричні інтеграли.
У всіх цих формулах під a розуміється ненульова константа , C означає сталу інтегрування .
синус
∫
sin
a
x
d
x
=
−
1
a
cos
a
x
+
C
{\displaystyle \int \sin ax\;dx=-{\frac {1}{a}\cos ax+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
{\displaystyle \int \sin ^{2}{ax}\;dx={\frac {x}{2}-{\frac {1}{4a}\sin 2ax+C={\frac {x}{2}-{\frac {1}{2a}\sin ax\cos ax+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
{\displaystyle \int x\sin ^{2}{ax}\;dx={\frac {x^{2}{4}-{\frac {x}{4a}\sin 2ax-{\frac {1}{8a^{2}\cos 2ax+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
{\displaystyle \int x^{2}\sin ^{2}{ax}\;dx={\frac {x^{3}{6}-\left({\frac {x^{2}{4a}-{\frac {1}{8a^{3}\right)\sin 2ax-{\frac {x}{4a^{2}\cos 2ax+C\!}
∫
sin
b
1
x
sin
b
2
x
d
x
=
sin
(
(
b
1
−
b
2
)
x
)
2
(
b
1
−
b
2
)
−
sin
(
(
b
1
+
b
2
)
x
)
2
(
b
1
+
b
2
)
+
C
(for
|
b
1
|
≠
|
b
2
|
)
{\displaystyle \int \sin b_{1}x\sin b_{2}x\;dx={\frac {\sin((b_{1}-b_{2})x)}{2(b_{1}-b_{2})}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}+C\qquad {\mbox{(for }|b_{1}|\neq |b_{2}|{\mbox{)}\,\!}
∫
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
)
{\displaystyle \int \sin ^{n}{ax}\;dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}+{\frac {n-1}{n}\int \sin ^{n-2}ax\;dx\qquad {\mbox{(for }n>0{\mbox{)}\,\!}
∫
d
x
sin
a
x
=
1
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {dx}{\sin ax}={\frac {1}{a}\ln \left|\tan {\frac {ax}{2}\right|+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
)
{\displaystyle \int {\frac {dx}{\sin ^{n}ax}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}+{\frac {n-2}{n-1}\int {\frac {dx}{\sin ^{n-2}ax}\qquad {\mbox{(for }n>1{\mbox{)}\,\!}
∫
x
sin
a
x
d
x
=
sin
a
x
a
2
−
x
cos
a
x
a
+
C
{\displaystyle \int x\sin ax\;dx={\frac {\sin ax}{a^{2}-{\frac {x\cos ax}{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
(for
n
>
0
)
{\displaystyle \int x^{n}\sin ax\;dx=-{\frac {x^{n}{a}\cos ax+{\frac {n}{a}\int x^{n-1}\cos ax\;dx\qquad {\mbox{(for }n>0{\mbox{)}\,\!}
∫
−
a
2
a
2
x
2
sin
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
)
24
n
2
π
2
(for
n
=
2
,
4
,
6...
)
{\displaystyle \int _{\frac {-a}{2}^{\frac {a}{2}x^{2}\sin ^{2}{\frac {n\pi x}{a}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}\qquad {\mbox{(for }n=2,4,6...{\mbox{)}\,\!}
∫
sin
a
x
x
d
x
=
∑
n
=
0
∞
(
−
1
)
n
(
a
x
)
2
n
+
1
(
2
n
+
1
)
⋅
(
2
n
+
1
)
!
+
C
{\displaystyle \int {\frac {\sin ax}{x}dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}{(2n+1)\cdot (2n+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
{\displaystyle \int {\frac {\sin ax}{x^{n}dx=-{\frac {\sin ax}{(n-1)x^{n-1}+{\frac {a}{n-1}\int {\frac {\cos ax}{x^{n-1}dx\,\!}
∫
d
x
1
±
sin
a
x
=
1
a
tan
(
a
x
2
∓
π
4
)
+
C
{\displaystyle \int {\frac {dx}{1\pm \sin ax}={\frac {1}{a}\tan \left({\frac {ax}{2}\mp {\frac {\pi }{4}\right)+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
{\displaystyle \int {\frac {x\;dx}{1+\sin ax}={\frac {x}{a}\tan \left({\frac {ax}{2}-{\frac {\pi }{4}\right)+{\frac {2}{a^{2}\ln \left|\cos \left({\frac {ax}{2}-{\frac {\pi }{4}\right)\right|+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
{\displaystyle \int {\frac {x\;dx}{1-\sin ax}={\frac {x}{a}\cot \left({\frac {\pi }{4}-{\frac {ax}{2}\right)+{\frac {2}{a^{2}\ln \left|\sin \left({\frac {\pi }{4}-{\frac {ax}{2}\right)\right|+C}
∫
sin
a
x
d
x
1
±
sin
a
x
=
±
x
+
1
a
tan
(
π
4
∓
a
x
2
)
+
C
{\displaystyle \int {\frac {\sin ax\;dx}{1\pm \sin ax}=\pm x+{\frac {1}{a}\tan \left({\frac {\pi }{4}\mp {\frac {ax}{2}\right)+C}
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
C
{\displaystyle \int \cos ax\;dx={\frac {1}{a}\sin ax+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
)
{\displaystyle \int \cos ^{n}ax\;dx={\frac {\cos ^{n-1}ax\sin ax}{na}+{\frac {n-1}{n}\int \cos ^{n-2}ax\;dx\qquad {\mbox{(for }n>0{\mbox{)}\,\!}
∫
x
cos
a
x
d
x
=
cos
a
x
a
2
+
x
sin
a
x
a
+
C
{\displaystyle \int x\cos ax\;dx={\frac {\cos ax}{a^{2}+{\frac {x\sin ax}{a}+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
{\displaystyle \int \cos ^{2}{ax}\;dx={\frac {x}{2}+{\frac {1}{4a}\sin 2ax+C={\frac {x}{2}+{\frac {1}{2a}\sin ax\cos ax+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
{\displaystyle \int x^{2}\cos ^{2}{ax}\;dx={\frac {x^{3}{6}+\left({\frac {x^{2}{4a}-{\frac {1}{8a^{3}\right)\sin 2ax+{\frac {x}{4a^{2}\cos 2ax+C\!}
∫
x
n
cos
a
x
d
x
=
x
n
sin
a
x
a
−
n
a
∫
x
n
−
1
sin
a
x
d
x
{\displaystyle \int x^{n}\cos ax\;dx={\frac {x^{n}\sin ax}{a}-{\frac {n}{a}\int x^{n-1}\sin ax\;dx\,\!}
∫
cos
a
x
x
d
x
=
ln
|
a
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
a
x
)
2
k
2
k
⋅
(
2
k
)
!
+
C
{\displaystyle \int {\frac {\cos ax}{x}dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}{2k\cdot (2k)!}+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
)
{\displaystyle \int {\frac {\cos ax}{x^{n}dx=-{\frac {\cos ax}{(n-1)x^{n-1}-{\frac {a}{n-1}\int {\frac {\sin ax}{x^{n-1}dx\qquad {\mbox{(for }n\neq 1{\mbox{)}\,\!}
∫
d
x
cos
a
x
=
1
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {dx}{\cos ax}={\frac {1}{a}\ln \left|\tan \left({\frac {ax}{2}+{\frac {\pi }{4}\right)\right|+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
)
{\displaystyle \int {\frac {dx}{\cos ^{n}ax}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}+{\frac {n-2}{n-1}\int {\frac {dx}{\cos ^{n-2}ax}\qquad {\mbox{(for }n>1{\mbox{)}\,\!}
∫
d
x
1
+
cos
a
x
=
1
a
tan
a
x
2
+
C
{\displaystyle \int {\frac {dx}{1+\cos ax}={\frac {1}{a}\tan {\frac {ax}{2}+C\,\!}
∫
d
x
1
−
cos
a
x
=
−
1
a
cot
a
x
2
+
C
{\displaystyle \int {\frac {dx}{1-\cos ax}=-{\frac {1}{a}\cot {\frac {ax}{2}+C\,\!}
∫
x
d
x
1
+
cos
a
x
=
x
a
tan
a
x
2
+
2
a
2
ln
|
cos
a
x
2
|
+
C
{\displaystyle \int {\frac {x\;dx}{1+\cos ax}={\frac {x}{a}\tan {\frac {ax}{2}+{\frac {2}{a^{2}\ln \left|\cos {\frac {ax}{2}\right|+C}
∫
x
d
x
1
−
cos
a
x
=
−
x
a
cot
a
x
2
+
2
a
2
ln
|
sin
a
x
2
|
+
C
{\displaystyle \int {\frac {x\;dx}{1-\cos ax}=-{\frac {x}{a}\cot {\frac {ax}{2}+{\frac {2}{a^{2}\ln \left|\sin {\frac {ax}{2}\right|+C}
∫
cos
a
x
d
x
1
+
cos
a
x
=
x
−
1
a
tan
a
x
2
+
C
{\displaystyle \int {\frac {\cos ax\;dx}{1+\cos ax}=x-{\frac {1}{a}\tan {\frac {ax}{2}+C\,\!}
∫
cos
a
x
d
x
1
−
cos
a
x
=
−
x
−
1
a
cot
a
x
2
+
C
{\displaystyle \int {\frac {\cos ax\;dx}{1-\cos ax}=-x-{\frac {1}{a}\cot {\frac {ax}{2}+C\,\!}
∫
cos
a
1
x
cos
a
2
x
d
x
=
sin
(
a
1
−
a
2
)
x
2
(
a
1
−
a
2
)
+
sin
(
a
1
+
a
2
)
x
2
(
a
1
+
a
2
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
{\displaystyle \int \cos a_{1}x\cos a_{2}x\;dx={\frac {\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}+{\frac {\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}+C\qquad {\mbox{(for }|a_{1}|\neq |a_{2}|{\mbox{)}\,\!}
∫
tan
a
x
d
x
=
−
1
a
ln
|
cos
a
x
|
+
C
=
1
a
ln
|
sec
a
x
|
+
C
{\displaystyle \int \tan ax\;dx=-{\frac {1}{a}\ln |\cos ax|+C={\frac {1}{a}\ln |\sec ax|+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
)
{\displaystyle \int \tan ^{n}ax\;dx={\frac {1}{a(n-1)}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;dx\qquad {\mbox{(for }n\neq 1{\mbox{)}\,\!}
∫
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
)
{\displaystyle \int {\frac {dx}{q\tan ax+p}={\frac {1}{p^{2}+q^{2}(px+{\frac {q}{a}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }p^{2}+q^{2}\neq 0{\mbox{)}\,\!}
∫
d
x
tan
a
x
=
1
a
ln
|
sin
a
x
|
+
C
{\displaystyle \int {\frac {dx}{\tan ax}={\frac {1}{a}\ln |\sin ax|+C\,\!}
∫
d
x
tan
a
x
+
1
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {dx}{\tan ax+1}={\frac {x}{2}+{\frac {1}{2a}\ln |\sin ax+\cos ax|+C\,\!}
∫
d
x
tan
a
x
−
1
=
−
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {dx}{\tan ax-1}=-{\frac {x}{2}+{\frac {1}{2a}\ln |\sin ax-\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
+
1
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax+1}={\frac {x}{2}-{\frac {1}{2a}\ln |\sin ax+\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
−
1
=
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax-1}={\frac {x}{2}+{\frac {1}{2a}\ln |\sin ax-\cos ax|+C\,\!}
секанс
∫
sec
a
x
d
x
=
1
a
ln
|
sec
a
x
+
tan
a
x
|
+
C
{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}
∫
sec
n
a
x
d
x
=
sec
n
−
1
a
x
sin
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
sec
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-1}{ax}\sin {ax}{a(n-1)}\,+\,{\frac {n-2}{n-1}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }n\neq 1{\mbox{)}\,\!}
∫
sec
n
x
d
x
=
sec
n
−
2
x
tan
x
n
−
1
+
n
−
2
n
−
1
∫
sec
n
−
2
x
d
x
{\displaystyle \int \sec ^{n}{x}\,dx={\frac {\sec ^{n-2}{x}\tan {x}{n-1}\,+\,{\frac {n-2}{n-1}\int \sec ^{n-2}{x}\,dx}
[ 1]
∫
d
x
sec
x
+
1
=
x
−
tan
x
2
+
C
{\displaystyle \int {\frac {dx}{\sec {x}+1}=x-\tan {\frac {x}{2}+C}
∫
d
x
sec
x
−
1
=
−
x
−
cot
x
2
+
C
{\displaystyle \int {\frac {dx}{\sec {x}-1}=-x-\cot {\frac {x}{2}+C}
∫
csc
a
x
d
x
=
1
a
ln
|
csc
a
x
−
cot
a
x
|
+
C
{\displaystyle \int \csc {ax}\,dx={\frac {1}{a}\ln {\left|\csc {ax}-\cot {ax}\right|}+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}
∫
csc
n
a
x
d
x
=
−
csc
n
−
1
a
x
cos
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
csc
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-1}{ax}\cos {ax}{a(n-1)}\,+\,{\frac {n-2}{n-1}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }n\neq 1{\mbox{)}\,\!}
∫
d
x
csc
x
+
1
=
x
−
2
sin
x
2
cos
x
2
+
sin
x
2
+
C
{\displaystyle \int {\frac {dx}{\csc {x}+1}=x-{\frac {2\sin {\frac {x}{2}{\cos {\frac {x}{2}+\sin {\frac {x}{2}+C}
∫
d
x
csc
x
−
1
=
2
sin
x
2
cos
x
2
−
sin
x
2
−
x
+
C
{\displaystyle \int {\frac {dx}{\csc {x}-1}={\frac {2\sin {\frac {x}{2}{\cos {\frac {x}{2}-\sin {\frac {x}{2}-x+C}
∫
cot
a
x
d
x
=
1
a
ln
|
sin
a
x
|
+
C
{\displaystyle \int \cot ax\;dx={\frac {1}{a}\ln |\sin ax|+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
)
{\displaystyle \int \cot ^{n}ax\;dx=-{\frac {1}{a(n-1)}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;dx\qquad {\mbox{(for }n\neq 1{\mbox{)}\,\!}
∫
d
x
1
+
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
+
1
{\displaystyle \int {\frac {dx}{1+\cot ax}=\int {\frac {\tan ax\;dx}{\tan ax+1}\,\!}
∫
d
x
1
−
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
−
1
{\displaystyle \int {\frac {dx}{1-\cot ax}=\int {\frac {\tan ax\;dx}{\tan ax-1}\,\!}
∫
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
n
−
1
(
sin
x
−
cos
x
(
cos
x
+
sin
x
)
n
−
1
−
2
(
n
−
2
)
∫
d
x
(
cos
x
+
sin
x
)
n
−
2
)
{\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}={\frac {1}{n-1}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}-2(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
)
{\displaystyle \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{)}\,\!}
також:
∫
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
)
{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;dx={\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{)}\,\!}
∫
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
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
)
{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}={\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}\qquad {\mbox{(for }m\neq 1{\mbox{)}\,\!}
також:
∫
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
)
{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}=-{\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}\qquad {\mbox{(for }m\neq n{\mbox{)}\,\!}
також:
∫
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
)
{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}={\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}\qquad {\mbox{(for }m\neq 1{\mbox{)}\,\!}
∫
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
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}=-{\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}\qquad {\mbox{(for }m\neq 1{\mbox{)}\,\!}
також:
∫
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
)
{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}={\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}\qquad {\mbox{(for }m\neq n{\mbox{)}\,\!}
також:
∫
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
)
{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}=-{\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}\qquad {\mbox{(for }m\neq 1{\mbox{)}\,\!}
∫
sin
a
x
tan
a
x
d
x
=
1
a
(
ln
|
sec
a
x
+
tan
a
x
|
−
sin
a
x
)
+
C
{\displaystyle \int \sin ax\tan ax\;dx={\frac {1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}
∫
tan
n
a
x
d
x
sin
2
a
x
=
1
a
(
n
−
1
)
tan
n
−
1
(
a
x
)
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\sin ^{2}ax}={\frac {1}{a(n-1)}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }n\neq 1{\mbox{)}\,\!}
∫
tan
n
a
x
d
x
cos
2
a
x
=
1
a
(
n
+
1
)
tan
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\cos ^{2}ax}={\frac {1}{a(n+1)}\tan ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)}\,\!}
∫
cot
n
a
x
d
x
sin
2
a
x
=
1
a
(
n
+
1
)
cot
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\sin ^{2}ax}={\frac {1}{a(n+1)}\cot ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)}\,\!}
∫
cot
n
a
x
d
x
cos
2
a
x
=
1
a
(
1
−
n
)
tan
1
−
n
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\cos ^{2}ax}={\frac {1}{a(1-n)}\tan ^{1-n}ax+C\qquad {\mbox{(for }n\neq 1{\mbox{)}\,\!}
∫
−
c
c
sin
x
d
x
=
0
{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}
∫
−
c
c
cos
x
d
x
=
2
∫
0
c
cos
x
d
x
=
2
∫
−
c
0
cos
x
d
x
=
2
sin
c
{\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx=2\sin {c}\!}
∫
−
c
c
tan
x
d
x
=
0
{\displaystyle \int _{-c}^{c}\tan {x}\;dx=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...
)
{\displaystyle \int _{\frac {-a}{2}^{\frac {a}{2}x^{2}\cos ^{2}{\frac {n\pi x}{a}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}\qquad {\mbox{(for }n=1,3,5...{\mbox{)}\,\!}
↑ Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008
Двайт Г. Б. Тригонометрические функции — интегралы // Таблицы интегралов и другие математические формулы / пер. с англ. Н. В. Леви ; под ред. К. А. Семендяева. — М . : Наука , 1978. — С. 83-102.(рос.)
The article is a derivative under the Creative Commons Attribution-ShareAlike License .
A link to the original article can be found here and attribution parties here
By using this site, you agree to the Terms of Use . Gpedia ® is a registered trademark of the Cyberajah Pty Ltd
