Материал из Википедии — свободной энциклопедии
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Ниже приведён список интегралов (первообразных функций) от обратных тригонометрических функций .
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
x
2
+
C
{\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}
∫
arcsin
x
a
d
x
=
x
arcsin
x
a
+
a
2
−
x
2
+
C
{\displaystyle \int \arcsin {\frac {x}{a}}\,dx=x\arcsin {\frac {x}{a}}+{\sqrt {a^{2}-x^{2}}}+C}
∫
x
arcsin
x
a
d
x
=
(
x
2
2
−
a
2
4
)
arcsin
x
a
+
x
4
a
2
−
x
2
+
C
{\displaystyle \int x\arcsin {\frac {x}{a}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}}{4}}\right)\arcsin {\frac {x}{a}}+{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C}
∫
x
2
arcsin
x
a
d
x
=
x
3
3
arcsin
x
a
+
x
2
+
2
a
2
9
a
2
−
x
2
+
C
{\displaystyle \int x^{2}\arcsin {\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{a}}+{\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C}
∫
x
n
arcsin
x
d
x
=
1
n
+
1
(
x
n
+
1
arcsin
x
+
x
n
1
−
x
2
−
n
x
n
−
1
arcsin
x
n
−
1
+
n
∫
x
n
−
2
arcsin
x
d
x
)
{\displaystyle \int x^{n}\arcsin x\,dx={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n-1}}+n\int x^{n-2}\arcsin x\,dx\right)}
∫
cos
n
x
arcsin
x
d
x
=
(
x
n
2
+
1
arccos
x
+
x
n
1
−
x
4
−
n
x
n
2
−
1
arccos
x
n
2
−
1
+
n
∫
x
n
2
−
2
arccos
x
d
x
)
{\displaystyle \int \cos ^{n}x\arcsin x\,dx=\left(x^{n^{2}+1}\arccos x+{\frac {x^{n}{\sqrt {1-x^{4}}}-nx^{n^{2}-1}\arccos x}{n^{2}-1}}+n\int x^{n^{2}-2}\arccos x\,dx\right)}
∫
arccos
x
d
x
=
x
arccos
x
−
1
−
x
2
+
C
{\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}
∫
arccos
x
a
d
x
=
x
arccos
x
a
−
a
2
−
x
2
+
C
{\displaystyle \int \arccos {\frac {x}{a}}\,dx=x\arccos {\frac {x}{a}}-{\sqrt {a^{2}-x^{2}}}+C}
∫
x
arccos
x
a
d
x
=
(
x
2
2
−
a
2
4
)
arccos
x
a
−
x
4
a
2
−
x
2
+
C
{\displaystyle \int x\arccos {\frac {x}{a}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}}{4}}\right)\arccos {\frac {x}{a}}-{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C}
∫
x
2
arccos
x
a
d
x
=
x
3
3
arccos
x
a
−
x
2
+
2
a
2
9
a
2
−
x
2
+
C
{\displaystyle \int x^{2}\arccos {\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\arccos {\frac {x}{a}}-{\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C}
∫
arctg
x
d
x
=
x
arctg
x
−
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \operatorname {arctg} \,x\,dx=x\,\operatorname {arctg} \,x-{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arctg
x
a
d
x
=
x
arctg
x
a
−
a
2
ln
(
1
+
x
2
a
2
)
+
C
{\displaystyle \int \operatorname {arctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}
∫
x
arctg
x
a
d
x
=
(
a
2
+
x
2
)
arctg
x
a
−
a
x
2
+
C
{\displaystyle \int x\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {(a^{2}+x^{2})\,\operatorname {arctg} \,{\frac {x}{a}}-ax}{2}}+C}
∫
x
2
arctg
x
a
d
x
=
x
3
3
arctg
x
a
−
a
x
2
6
+
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}
∫
x
n
arctg
x
a
d
x
=
x
n
+
1
n
+
1
arctg
x
a
−
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1}}\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\,dx,\quad n\neq -1}
∫
arcctg
x
d
x
=
x
arcctg
x
+
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \operatorname {arcctg} \,x\,dx=x\,\operatorname {arcctg} \,x+{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arcctg
x
a
d
x
=
x
arcctg
x
a
+
a
2
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arcctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}
∫
x
arcctg
x
a
d
x
=
a
2
+
x
2
2
arcctg
x
a
+
a
x
2
+
C
{\displaystyle \int x\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {a^{2}+x^{2}}{2}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {ax}{2}}+C}
∫
x
2
arcctg
x
a
d
x
=
x
3
3
arcctg
x
a
+
a
x
2
6
−
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}
∫
x
n
arcctg
x
a
d
x
=
x
n
+
1
n
+
1
arcctg
x
a
+
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\,dx,\quad n\neq -1}
∫
arcsec
x
d
x
=
x
arcsec
x
−
ln
|
x
+
x
x
2
−
1
x
2
|
+
C
{\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫
arcsec
x
a
d
x
=
x
arcsec
x
a
+
x
a
|
x
|
ln
|
x
±
x
2
−
1
|
+
C
{\displaystyle \int \operatorname {arcsec} {\frac {x}{a}}\,dx=x\operatorname {arcsec} {\frac {x}{a}}+{\frac {x}{a|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|+C}
∫
x
arcsec
x
d
x
=
1
2
(
x
2
arcsec
x
−
x
2
−
1
)
+
C
{\displaystyle \int x\operatorname {arcsec} x\,dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)+C}
∫
x
n
arcsec
x
d
x
=
1
n
+
1
(
x
n
+
1
arcsec
x
−
1
n
[
x
n
−
1
x
2
−
1
+
[
1
−
n
]
(
x
n
−
1
arcsec
x
+
(
1
−
n
)
∫
x
n
−
2
arcsec
x
d
x
)
]
)
{\displaystyle \int x^{n}\operatorname {arcsec} x\,dx={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+[1-n]\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\,dx\right)\right]\right)}
∫
arccosec
x
d
x
=
x
arccosec
x
+
ln
|
x
+
x
x
2
−
1
x
2
|
+
C
{\displaystyle \int \operatorname {arccosec} \,x\,dx=x\,\operatorname {arccosec} \,x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫
arccosec
x
a
d
x
=
x
arccosec
x
a
+
a
ln
(
x
a
(
1
−
a
2
x
2
+
1
)
)
+
C
{\displaystyle \int \,\operatorname {arccosec} \,{\frac {x}{a}}\,dx=x\,\operatorname {arccosec} \,{\frac {x}{a}}+{a}\ln {\left({\frac {x}{a}}\left({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1\right)\right)}+C}
∫
x
arccosec
x
a
d
x
=
x
2
2
arccosec
x
a
+
a
x
2
1
−
a
2
x
2
+
C
{\displaystyle \int x\,\operatorname {arccosec} \,{\frac {x}{a}}\,dx={\frac {x^{2}}{2}}\,\operatorname {arccosec} \,{\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}
Книги Таблицы интегралов Вычисление интегралов
Списки интегралов по типам функций
![{\displaystyle \int x^{n}\operatorname {arcsec} x\,dx={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+[1-n]\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\,dx\right)\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df6b1b242f587f4c7d8f44f63fddc972c62ba510)
