Список интегралов от тригонометрических функций
Ниже приведён список интегралов (первообразных функций) от тригонометрических функций . В списке везде опущена аддитивная константа интегрирования.
Константа
c
{\displaystyle c}
не равняется нулю.
Интегралы, содержащие только синус
∫
sin
c
x
d
x
=
−
1
c
cos
c
x
+
C
{\displaystyle \int \sin cx\;dx=-{\frac {1}{c}\cos cx+C}
∫
sin
n
c
x
d
x
=
−
sin
n
−
1
c
x
cos
c
x
n
c
+
n
−
1
n
∫
sin
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \sin ^{n}cx\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}+{\frac {n-1}{n}\int \sin ^{n-2}cx\;dx\qquad {\mbox{( }n>0{\mbox{)}
∫
x
sin
c
x
d
x
=
sin
c
x
c
2
−
x
cos
c
x
c
{\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}-{\frac {x\cos cx}{c}
∫
x
2
sin
c
x
d
x
=
2
cos
c
x
c
3
+
2
x
sin
c
x
c
2
−
x
2
cos
c
x
c
{\displaystyle \int x^{2}\sin cx\;dx={\frac {2\cos cx}{c^{3}+{\frac {2x\sin cx}{c^{2}-{\frac {x^{2}\cos cx}{c}
∫
x
3
sin
c
x
d
x
=
−
6
sin
c
x
c
4
+
6
x
cos
c
x
c
3
+
3
x
2
sin
c
x
c
2
−
x
3
cos
c
x
c
{\displaystyle \int x^{3}\sin cx\;dx=-{\frac {6\sin cx}{c^{4}+{\frac {6x\cos cx}{c^{3}+{\frac {3x^{2}\sin cx}{c^{2}-{\frac {x^{3}\cos cx}{c}
∫
x
4
sin
c
x
d
x
=
−
24
cos
c
x
c
5
−
24
x
sin
c
x
c
4
+
12
x
2
cos
c
x
c
3
+
4
x
3
sin
c
x
c
2
−
x
4
cos
c
x
c
{\displaystyle \int x^{4}\sin cx\;dx=-{\frac {24\cos cx}{c^{5}-{\frac {24x\sin cx}{c^{4}+{\frac {12x^{2}\cos cx}{c^{3}+{\frac {4x^{3}\sin cx}{c^{2}-{\frac {x^{4}\cos cx}{c}
∫
x
5
sin
c
x
d
x
=
120
sin
c
x
c
6
−
120
x
cos
c
x
c
5
−
60
x
2
sin
c
x
c
4
+
20
x
3
cos
c
x
c
3
+
5
x
4
sin
c
x
c
2
−
x
5
cos
c
x
c
{\displaystyle \int x^{5}\sin cx\;dx={\frac {120\sin cx}{c^{6}-{\frac {120x\cos cx}{c^{5}-{\frac {60x^{2}\sin cx}{c^{4}+{\frac {20x^{3}\cos cx}{c^{3}+{\frac {5x^{4}\sin cx}{c^{2}-{\frac {x^{5}\cos cx}{c}
∫
x
n
sin
c
x
d
x
=
n
!
⋅
sin
c
x
[
x
n
−
1
c
2
⋅
(
n
−
1
)
!
−
x
n
−
3
c
4
⋅
(
n
−
3
)
!
+
x
n
−
5
c
6
⋅
(
n
−
5
)
!
−
.
.
.
]
−
−
n
!
⋅
cos
c
x
[
x
n
c
⋅
n
!
−
x
n
−
2
c
3
⋅
(
n
−
2
)
!
+
x
n
−
4
c
5
⋅
(
n
−
4
)
!
−
.
.
.
]
{\displaystyle {\begin{aligned}\int x^{n}\sin cx\;dx&=n!\cdot \sin cx\left[{\frac {x^{n-1}{c^{2}\cdot (n-1)!}-{\frac {x^{n-3}{c^{4}\cdot (n-3)!}+{\frac {x^{n-5}{c^{6}\cdot (n-5)!}-...\right]-\\&-n!\cdot \cos cx\left[{\frac {x^{n}{c\cdot n!}-{\frac {x^{n-2}{c^{3}\cdot (n-2)!}+{\frac {x^{n-4}{c^{5}\cdot (n-4)!}-...\right]\end{aligned}
∫
x
n
sin
c
x
d
x
=
−
x
n
c
cos
c
x
+
n
c
∫
x
n
−
1
cos
c
x
d
x
(
n
≥
0
)
{\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}{c}\cos cx+{\frac {n}{c}\int x^{n-1}\cos cx\;dx\qquad {\mbox{( }n\geq 0{\mbox{)}
∫
sin
c
x
x
d
x
=
∑
i
=
0
∞
(
−
1
)
i
(
c
x
)
2
i
+
1
(
2
i
+
1
)
⋅
(
2
i
+
1
)
!
{\displaystyle \int {\frac {\sin cx}{x}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}{(2i+1)\cdot (2i+1)!}
∫
sin
c
x
x
n
d
x
=
−
sin
c
x
(
n
−
1
)
x
n
−
1
+
c
n
−
1
∫
cos
c
x
x
n
−
1
d
x
{\displaystyle \int {\frac {\sin cx}{x^{n}dx=-{\frac {\sin cx}{(n-1)x^{n-1}+{\frac {c}{n-1}\int {\frac {\cos cx}{x^{n-1}dx}
∫
d
x
sin
c
x
=
1
c
ln
|
tg
c
x
2
|
{\displaystyle \int {\frac {dx}{\sin cx}={\frac {1}{c}\ln \left|\operatorname {tg} {\frac {cx}{2}\right|}
∫
d
x
sin
n
c
x
=
cos
c
x
c
(
1
−
n
)
sin
n
−
1
c
x
+
n
−
2
n
−
1
∫
d
x
sin
n
−
2
c
x
(
n
>
1
)
{\displaystyle \int {\frac {dx}{\sin ^{n}cx}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}+{\frac {n-2}{n-1}\int {\frac {dx}{\sin ^{n-2}cx}\qquad {\mbox{( }n>1{\mbox{)}
∫
d
x
1
±
sin
c
x
=
1
c
tg
(
c
x
2
∓
π
4
)
{\displaystyle \int {\frac {dx}{1\pm \sin cx}={\frac {1}{c}\operatorname {tg} \left({\frac {cx}{2}\mp {\frac {\pi }{4}\right)}
∫
x
d
x
1
+
sin
c
x
=
x
c
tg
(
c
x
2
−
π
4
)
+
2
c
2
ln
|
cos
(
c
x
2
−
π
4
)
|
{\displaystyle \int {\frac {x\;dx}{1+\sin cx}={\frac {x}{c}\operatorname {tg} \left({\frac {cx}{2}-{\frac {\pi }{4}\right)+{\frac {2}{c^{2}\ln \left|\cos \left({\frac {cx}{2}-{\frac {\pi }{4}\right)\right|}
∫
x
d
x
1
−
sin
c
x
=
x
c
ctg
(
π
4
−
c
x
2
)
+
2
c
2
ln
|
sin
(
π
4
−
c
x
2
)
|
{\displaystyle \int {\frac {x\;dx}{1-\sin cx}={\frac {x}{c}\operatorname {ctg} \left({\frac {\pi }{4}-{\frac {cx}{2}\right)+{\frac {2}{c^{2}\ln \left|\sin \left({\frac {\pi }{4}-{\frac {cx}{2}\right)\right|}
∫
sin
c
x
d
x
1
±
sin
c
x
=
±
x
+
1
c
tg
(
π
4
∓
c
x
2
)
{\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}=\pm x+{\frac {1}{c}\operatorname {tg} \left({\frac {\pi }{4}\mp {\frac {cx}{2}\right)}
∫
sin
c
1
x
sin
c
2
x
d
x
=
sin
(
(
c
1
−
c
2
)
x
)
2
(
c
1
−
c
2
)
−
sin
(
(
c
1
+
c
2
)
x
)
2
(
c
1
+
c
2
)
(
|
c
1
|
≠
|
c
2
|
)
{\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin((c_{1}-c_{2})x)}{2(c_{1}-c_{2})}-{\frac {\sin((c_{1}+c_{2})x)}{2(c_{1}+c_{2})}\qquad {\mbox{( }|c_{1}|\neq |c_{2}|{\mbox{)}
Интегралы, содержащие только косинус
∫
cos
c
x
d
x
=
1
c
sin
c
x
+
C
{\displaystyle \int \cos cx\;dx={\frac {1}{c}\sin cx+C}
∫
cos
n
c
x
d
x
=
cos
n
−
1
c
x
sin
c
x
n
c
+
n
−
1
n
∫
cos
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \cos ^{n}cx\;dx={\frac {\cos ^{n-1}cx\sin cx}{nc}+{\frac {n-1}{n}\int \cos ^{n-2}cx\;dx\qquad {\mbox{( }n>0{\mbox{)}
∫
x
cos
c
x
d
x
=
cos
c
x
c
2
+
x
sin
c
x
c
{\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}+{\frac {x\sin cx}{c}
∫
x
n
cos
c
x
d
x
=
x
n
sin
c
x
c
−
n
c
∫
x
n
−
1
sin
c
x
d
x
{\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}-{\frac {n}{c}\int x^{n-1}\sin cx\;dx}
∫
cos
c
x
x
d
x
=
ln
|
c
x
|
+
∑
i
=
1
∞
(
−
1
)
i
(
c
x
)
2
i
2
i
⋅
(
2
i
)
!
{\displaystyle \int {\frac {\cos cx}{x}dx=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}{2i\cdot (2i)!}
∫
cos
c
x
x
n
d
x
=
−
cos
c
x
(
n
−
1
)
x
n
−
1
−
c
n
−
1
∫
sin
c
x
x
n
−
1
d
x
(
n
≠
1
)
{\displaystyle \int {\frac {\cos cx}{x^{n}dx=-{\frac {\cos cx}{(n-1)x^{n-1}-{\frac {c}{n-1}\int {\frac {\sin cx}{x^{n-1}dx\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
d
x
cos
c
x
=
1
c
ln
|
tg
(
c
x
2
+
π
4
)
|
{\displaystyle \int {\frac {dx}{\cos cx}={\frac {1}{c}\ln \left|\operatorname {tg} \left({\frac {cx}{2}+{\frac {\pi }{4}\right)\right|}
∫
d
x
cos
n
c
x
=
sin
c
x
c
(
n
−
1
)
cos
n
−
1
c
x
+
n
−
2
n
−
1
∫
d
x
cos
n
−
2
c
x
(
n
>
1
)
{\displaystyle \int {\frac {dx}{\cos ^{n}cx}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}+{\frac {n-2}{n-1}\int {\frac {dx}{\cos ^{n-2}cx}\qquad {\mbox{( }n>1{\mbox{)}
∫
d
x
1
+
cos
c
x
=
1
c
tg
c
x
2
{\displaystyle \int {\frac {dx}{1+\cos cx}={\frac {1}{c}\operatorname {tg} {\frac {cx}{2}
∫
d
x
1
−
cos
c
x
=
−
1
c
ctg
c
x
2
{\displaystyle \int {\frac {dx}{1-\cos cx}=-{\frac {1}{c}\operatorname {ctg} {\frac {cx}{2}
∫
x
d
x
1
+
cos
c
x
=
x
c
tg
c
x
2
+
2
c
2
ln
|
cos
c
x
2
|
{\displaystyle \int {\frac {x\;dx}{1+\cos cx}={\frac {x}{c}\operatorname {tg} {\frac {cx}{2}+{\frac {2}{c^{2}\ln \left|\cos {\frac {cx}{2}\right|}
∫
x
d
x
1
−
cos
c
x
=
−
x
c
ctg
c
x
2
+
2
c
2
ln
|
sin
c
x
2
|
{\displaystyle \int {\frac {x\;dx}{1-\cos cx}=-{\frac {x}{c}\operatorname {ctg} {\frac {cx}{2}+{\frac {2}{c^{2}\ln \left|\sin {\frac {cx}{2}\right|}
∫
cos
c
x
d
x
1
+
cos
c
x
=
x
−
1
c
tg
c
x
2
{\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}=x-{\frac {1}{c}\operatorname {tg} {\frac {cx}{2}
∫
cos
c
x
d
x
1
−
cos
c
x
=
−
x
−
1
c
ctg
c
x
2
{\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}=-x-{\frac {1}{c}\operatorname {ctg} {\frac {cx}{2}
∫
cos
c
1
x
cos
c
2
x
d
x
=
sin
(
c
1
−
c
2
)
x
2
(
c
1
−
c
2
)
+
sin
(
c
1
+
c
2
)
x
2
(
c
1
+
c
2
)
(
|
c
1
|
≠
|
c
2
|
)
{\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}\qquad {\mbox{( }|c_{1}|\neq |c_{2}|{\mbox{)}
Интегралы, содержащие только тангенс
∫
tg
c
x
d
x
=
−
1
c
ln
|
cos
c
x
|
{\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}\ln |\cos cx|}
∫
tg
n
c
x
d
x
=
1
c
(
n
−
1
)
tg
n
−
1
c
x
−
∫
tg
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \operatorname {tg} ^{n}cx\;dx={\frac {1}{c(n-1)}\operatorname {tg} ^{n-1}cx-\int \operatorname {tg} ^{n-2}cx\;dx\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
d
x
tg
c
x
+
1
=
x
2
+
1
2
c
ln
|
sin
c
x
+
cos
c
x
|
{\displaystyle \int {\frac {dx}{\operatorname {tg} cx+1}={\frac {x}{2}+{\frac {1}{2c}\ln |\sin cx+\cos cx|}
∫
d
x
tg
c
x
−
1
=
−
x
2
+
1
2
c
ln
|
sin
c
x
−
cos
c
x
|
{\displaystyle \int {\frac {dx}{\operatorname {tg} cx-1}=-{\frac {x}{2}+{\frac {1}{2c}\ln |\sin cx-\cos cx|}
∫
tg
c
x
d
x
tg
c
x
+
1
=
x
2
−
1
2
c
ln
|
sin
c
x
+
cos
c
x
|
{\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}={\frac {x}{2}-{\frac {1}{2c}\ln |\sin cx+\cos cx|}
∫
tg
c
x
d
x
tg
c
x
−
1
=
x
2
+
1
2
c
ln
|
sin
c
x
−
cos
c
x
|
{\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}={\frac {x}{2}+{\frac {1}{2c}\ln |\sin cx-\cos cx|}
Интегралы, содержащие только секанс
∫
sec
c
x
d
x
=
1
c
ln
|
sec
c
x
+
tg
c
x
|
{\displaystyle \int \sec {cx}\,dx={\frac {1}{c}\ln {\left|\sec {cx}+\operatorname {tg} {cx}\right|}
∫
sec
n
c
x
d
x
=
sec
n
−
1
c
x
sin
c
x
c
(
n
−
1
)
+
n
−
2
n
−
1
∫
sec
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \sec ^{n}{cx}\,dx={\frac {\sec ^{n-1}{cx}\sin {cx}{c(n-1)}\,+\,{\frac {n-2}{n-1}\int \sec ^{n-2}{cx}\,dx\qquad {\mbox{ ( }n\neq 1{\mbox{)}
∫
d
x
sec
x
+
1
=
x
−
tg
x
2
{\displaystyle \int {\frac {dx}{\sec {x}+1}=x-\operatorname {tg} {\frac {x}{2}
∫
cosec
c
x
d
x
=
−
1
c
ln
|
cosec
c
x
+
ctg
c
x
|
{\displaystyle \int \operatorname {cosec} {cx}\,dx=-{\frac {1}{c}\ln {\left|\operatorname {cosec} {cx}+\operatorname {ctg} {cx}\right|}
∫
cosec
n
c
x
d
x
=
−
cosec
n
−
1
c
x
cos
c
x
c
(
n
−
1
)
+
n
−
2
n
−
1
∫
cosec
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \operatorname {cosec} ^{n}{cx}\,dx=-{\frac {\operatorname {cosec} ^{n-1}{cx}\cos {cx}{c(n-1)}\,+\,{\frac {n-2}{n-1}\int \operatorname {cosec} ^{n-2}{cx}\,dx\qquad {\mbox{ ( }n\neq 1{\mbox{)}
∫
ctg
c
x
d
x
=
1
c
ln
|
sin
c
x
|
{\displaystyle \int \operatorname {ctg} cx\;dx={\frac {1}{c}\ln |\sin cx|}
∫
ctg
n
c
x
d
x
=
−
1
c
(
n
−
1
)
ctg
n
−
1
c
x
−
∫
ctg
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \operatorname {ctg} ^{n}cx\;dx=-{\frac {1}{c(n-1)}\operatorname {ctg} ^{n-1}cx-\int \operatorname {ctg} ^{n-2}cx\;dx\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
d
x
1
+
ctg
c
x
=
∫
tg
c
x
d
x
tg
c
x
+
1
{\displaystyle \int {\frac {dx}{1+\operatorname {ctg} cx}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}
∫
d
x
1
−
ctg
c
x
=
∫
tg
c
x
d
x
tg
c
x
−
1
{\displaystyle \int {\frac {dx}{1-\operatorname {ctg} cx}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}
Интегралы, содержащие только синус и косинус
∫
d
x
cos
c
x
±
sin
c
x
=
1
c
2
ln
|
tg
(
c
x
2
±
π
8
)
|
{\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}={\frac {1}{c{\sqrt {2}\ln \left|\operatorname {tg} \left({\frac {cx}{2}\pm {\frac {\pi }{8}\right)\right|}
∫
d
x
(
cos
c
x
±
sin
c
x
)
2
=
1
2
c
tg
(
c
x
∓
π
4
)
{\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}={\frac {1}{2c}\operatorname {tg} \left(cx\mp {\frac {\pi }{4}\right)}
∫
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
c
x
d
x
cos
c
x
+
sin
c
x
=
x
2
+
1
2
c
ln
|
sin
c
x
+
cos
c
x
|
{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}={\frac {x}{2}+{\frac {1}{2c}\ln \left|\sin cx+\cos cx\right|}
∫
cos
c
x
d
x
cos
c
x
−
sin
c
x
=
x
2
−
1
2
c
ln
|
sin
c
x
−
cos
c
x
|
{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}={\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx-\cos cx\right|}
∫
sin
c
x
d
x
cos
c
x
+
sin
c
x
=
x
2
−
1
2
c
ln
|
sin
c
x
+
cos
c
x
|
{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}={\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx+\cos cx\right|}
∫
sin
c
x
d
x
cos
c
x
−
sin
c
x
=
−
x
2
−
1
2
c
ln
|
sin
c
x
−
cos
c
x
|
{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}=-{\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx-\cos cx\right|}
∫
cos
c
x
d
x
sin
c
x
(
1
+
cos
c
x
)
=
−
1
4
c
tg
2
c
x
2
+
1
2
c
ln
|
tg
c
x
2
|
{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}=-{\frac {1}{4c}\operatorname {tg} ^{2}{\frac {cx}{2}+{\frac {1}{2c}\ln \left|\operatorname {tg} {\frac {cx}{2}\right|}
∫
cos
c
x
d
x
sin
c
x
(
1
−
cos
c
x
)
=
−
1
4
c
ctg
2
c
x
2
−
1
2
c
ln
|
tg
c
x
2
|
{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1-\cos cx)}=-{\frac {1}{4c}\operatorname {ctg} ^{2}{\frac {cx}{2}-{\frac {1}{2c}\ln \left|\operatorname {tg} {\frac {cx}{2}\right|}
∫
sin
c
x
d
x
cos
c
x
(
1
+
sin
c
x
)
=
1
4
c
ctg
2
(
c
x
2
+
π
4
)
+
1
2
c
ln
|
tg
(
c
x
2
+
π
4
)
|
{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1+\sin cx)}={\frac {1}{4c}\operatorname {ctg} ^{2}\left({\frac {cx}{2}+{\frac {\pi }{4}\right)+{\frac {1}{2c}\ln \left|\operatorname {tg} \left({\frac {cx}{2}+{\frac {\pi }{4}\right)\right|}
∫
sin
c
x
d
x
cos
c
x
(
1
−
sin
c
x
)
=
1
4
c
tg
2
(
c
x
2
+
π
4
)
−
1
2
c
ln
|
tg
(
c
x
2
+
π
4
)
|
{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1-\sin cx)}={\frac {1}{4c}\operatorname {tg} ^{2}\left({\frac {cx}{2}+{\frac {\pi }{4}\right)-{\frac {1}{2c}\ln \left|\operatorname {tg} \left({\frac {cx}{2}+{\frac {\pi }{4}\right)\right|}
∫
sin
c
x
cos
c
x
d
x
=
1
2
c
sin
2
c
x
{\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}\sin ^{2}cx}
∫
sin
c
1
x
cos
c
2
x
d
x
=
−
cos
(
c
1
+
c
2
)
x
2
(
c
1
+
c
2
)
−
cos
(
c
1
−
c
2
)
x
2
(
c
1
−
c
2
)
(
|
c
1
|
≠
|
c
2
|
)
{\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}\qquad {\mbox{( }|c_{1}|\neq |c_{2}|{\mbox{)}
∫
sin
n
c
x
cos
c
x
d
x
=
1
c
(
n
+
1
)
sin
n
+
1
c
x
(
n
≠
1
)
{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}\sin ^{n+1}cx\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
c
x
cos
n
c
x
d
x
=
−
1
c
(
n
+
1
)
cos
n
+
1
c
x
(
n
≠
1
)
{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}\cos ^{n+1}cx\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
n
c
x
cos
m
c
x
d
x
=
−
sin
n
−
1
c
x
cos
m
+
1
c
x
c
(
n
+
m
)
+
n
−
1
n
+
m
∫
sin
n
−
2
c
x
cos
m
c
x
d
x
(
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}+{\frac {n-1}{n+m}\int \sin ^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{( }m,n>0{\mbox{)}
∫
sin
n
c
x
cos
m
c
x
d
x
=
sin
n
+
1
c
x
cos
m
−
1
c
x
c
(
n
+
m
)
+
m
−
1
n
+
m
∫
sin
n
c
x
cos
m
−
2
c
x
d
x
(
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}+{\frac {m-1}{n+m}\int \sin ^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{( }m,n>0{\mbox{)}
∫
d
x
sin
c
x
cos
c
x
=
1
c
ln
|
tg
c
x
|
{\displaystyle \int {\frac {dx}{\sin cx\cos cx}={\frac {1}{c}\ln \left|\operatorname {tg} cx\right|}
∫
d
x
sin
c
x
cos
n
c
x
=
1
c
(
n
−
1
)
cos
n
−
1
c
x
+
∫
d
x
sin
c
x
cos
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}={\frac {1}{c(n-1)\cos ^{n-1}cx}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
d
x
sin
n
c
x
cos
c
x
=
−
1
c
(
n
−
1
)
sin
n
−
1
c
x
+
∫
d
x
sin
n
−
2
c
x
cos
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
c
x
d
x
cos
n
c
x
=
1
c
(
n
−
1
)
cos
n
−
1
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}={\frac {1}{c(n-1)\cos ^{n-1}cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
2
c
x
d
x
cos
c
x
=
−
1
c
sin
c
x
+
1
c
ln
|
tg
(
π
4
+
c
x
2
)
|
{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos cx}=-{\frac {1}{c}\sin cx+{\frac {1}{c}\ln \left|\operatorname {tg} \left({\frac {\pi }{4}+{\frac {cx}{2}\right)\right|}
∫
sin
2
c
x
d
x
cos
n
c
x
=
sin
c
x
c
(
n
−
1
)
cos
n
−
1
c
x
−
1
n
−
1
∫
d
x
cos
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}-{\frac {1}{n-1}\int {\frac {dx}{\cos ^{n-2}cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
n
c
x
d
x
cos
c
x
=
−
sin
n
−
1
c
x
c
(
n
−
1
)
+
∫
sin
n
−
2
c
x
d
x
cos
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}=-{\frac {\sin ^{n-1}cx}{c(n-1)}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
sin
n
c
x
d
x
cos
m
c
x
=
sin
n
+
1
c
x
c
(
m
−
1
)
cos
m
−
1
c
x
−
n
−
m
+
2
m
−
1
∫
sin
n
c
x
d
x
cos
m
−
2
c
x
(
m
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}-{\frac {n-m+2}{m-1}\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m-2}cx}\qquad {\mbox{( }m\neq 1{\mbox{)}
∫
sin
n
c
x
d
x
cos
m
c
x
=
−
sin
n
−
1
c
x
c
(
n
−
m
)
cos
m
−
1
c
x
+
n
−
1
n
−
m
∫
sin
n
−
2
c
x
d
x
cos
m
c
x
(
m
≠
n
)
{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}+{\frac {n-1}{n-m}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m}cx}\qquad {\mbox{( }m\neq n{\mbox{)}
∫
sin
n
c
x
d
x
cos
m
c
x
=
sin
n
−
1
c
x
c
(
m
−
1
)
cos
m
−
1
c
x
−
n
−
1
m
−
1
∫
sin
n
−
1
c
x
d
x
cos
m
−
2
c
x
(
m
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}-{\frac {n-1}{m-1}\int {\frac {\sin ^{n-1}cx\;dx}{\cos ^{m-2}cx}\qquad {\mbox{( }m\neq 1{\mbox{)}
∫
cos
c
x
d
x
sin
n
c
x
=
−
1
c
(
n
−
1
)
sin
n
−
1
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
cos
2
c
x
d
x
sin
c
x
=
1
c
(
cos
c
x
+
ln
|
tg
c
x
2
|
)
{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}={\frac {1}{c}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}\right|\right)}
∫
cos
2
c
x
d
x
sin
n
c
x
=
−
1
n
−
1
(
cos
c
x
c
sin
n
−
1
c
x
)
+
∫
d
x
sin
n
−
2
c
x
)
(
n
≠
1
)
{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}=-{\frac {1}{n-1}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}+\int {\frac {dx}{\sin ^{n-2}cx}\right)\qquad {\mbox{( }n\neq 1{\mbox{)}
∫
cos
n
c
x
d
x
sin
m
c
x
=
−
cos
n
+
1
c
x
c
(
m
−
1
)
sin
m
−
1
c
x
−
n
−
m
−
2
m
−
1
∫
c
o
s
n
c
x
d
x
sin
m
−
2
c
x
(
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}-{\frac {n-m-2}{m-1}\int {\frac {cos^{n}cx\;dx}{\sin ^{m-2}cx}\qquad {\mbox{( }m\neq 1{\mbox{)}
∫
cos
n
c
x
d
x
sin
m
c
x
=
cos
n
−
1
c
x
c
(
n
−
m
)
sin
m
−
1
c
x
+
n
−
1
n
−
m
∫
c
o
s
n
−
2
c
x
d
x
sin
m
c
x
(
m
≠
n
)
{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}+{\frac {n-1}{n-m}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m}cx}\qquad {\mbox{( }m\neq n{\mbox{)}
∫
cos
n
c
x
d
x
sin
m
c
x
=
−
cos
n
−
1
c
x
c
(
m
−
1
)
sin
m
−
1
c
x
−
n
−
1
m
−
1
∫
c
o
s
n
−
2
c
x
d
x
sin
m
−
2
c
x
(
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}-{\frac {n-1}{m-1}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m-2}cx}\qquad {\mbox{( }m\neq 1{\mbox{)}
Интегралы, содержащие только синус и тангенс
∫
sin
c
x
tg
c
x
d
x
=
1
c
(
ln
|
sec
c
x
+
tg
c
x
|
−
sin
c
x
)
{\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)}
∫
tg
n
c
x
d
x
sin
2
c
x
=
1
c
(
n
−
1
)
tg
n
−
1
(
c
x
)
(
n
≠
1
)
{\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\sin ^{2}cx}={\frac {1}{c(n-1)}\operatorname {tg} ^{n-1}(cx)\qquad {\mbox{( }n\neq 1{\mbox{)}
Интегралы, содержащие только косинус и тангенс
∫
tg
n
c
x
d
x
cos
2
c
x
=
1
c
(
n
+
1
)
tg
n
+
1
c
x
(
n
≠
−
1
)
{\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\cos ^{2}cx}={\frac {1}{c(n+1)}\operatorname {tg} ^{n+1}cx\qquad {\mbox{( }n\neq -1{\mbox{)}
Интегралы, содержащие только синус и котангенс
∫
ctg
n
c
x
d
x
sin
2
c
x
=
1
c
(
n
+
1
)
ctg
n
+
1
c
x
(
n
≠
−
1
)
{\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\sin ^{2}cx}={\frac {1}{c(n+1)}\operatorname {ctg} ^{n+1}cx\qquad {\mbox{( }n\neq -1{\mbox{)}
Интегралы, содержащие только косинус и котангенс
∫
ctg
n
c
x
d
x
cos
2
c
x
=
1
c
(
1
−
n
)
tg
1
−
n
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\cos ^{2}cx}={\frac {1}{c(1-n)}\operatorname {tg} ^{1-n}cx\qquad {\mbox{( }n\neq 1{\mbox{)}
Интегралы, содержащие только тангенс и котангенс
∫
tg
m
(
c
x
)
ctg
n
(
c
x
)
d
x
=
1
c
(
m
+
n
−
1
)
tg
m
+
n
−
1
(
c
x
)
−
∫
tg
m
−
2
(
c
x
)
ctg
n
(
c
x
)
d
x
(
m
+
n
≠
1
)
{\displaystyle \int {\frac {\operatorname {tg} ^{m}(cx)}{\operatorname {ctg} ^{n}(cx)}\;dx={\frac {1}{c(m+n-1)}\operatorname {tg} ^{m+n-1}(cx)-\int {\frac {\operatorname {tg} ^{m-2}(cx)}{\operatorname {ctg} ^{n}(cx)}\;dx\qquad {\mbox{( }m+n\neq 1{\mbox{)}
Библиография
Книги
Таблицы интегралов
Вычисление интегралов
Списки интегралов по типам функций
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