==Arcsin
∫
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)}
∫
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}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arctan
(
x
a
)
d
x
=
x
arctan
(
x
a
)
−
a
2
ln
(
1
+
x
2
a
2
)
+
C
{\displaystyle \int \arctan {\big (}{\frac {x}{a}}{\big )}dx=x\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}
∫
x
arctan
(
x
a
)
d
x
=
(
a
2
+
x
2
)
arctan
(
x
a
)
−
a
x
2
+
C
{\displaystyle \int x\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {(a^{2}+x^{2})\arctan {\big (}{\frac {x}{a}}{\big )}-ax}{2}}+C}
∫
x
2
arctan
(
x
a
)
d
x
=
x
3
3
arctan
(
x
a
)
−
a
x
2
6
+
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}
∫
x
n
arctan
(
x
a
)
d
x
=
x
n
+
1
n
+
1
arctan
(
x
a
)
−
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
|
x
+
x
x
2
−
1
x
2
|
+
C
{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫
arccsc
x
a
d
x
=
x
arccsc
x
a
+
a
ln
(
x
a
(
1
−
a
2
x
2
+
1
)
)
+
C
{\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ dx=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {({\frac {x}{a}}({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1))}+C}
∫
x
arccsc
x
a
d
x
=
x
2
2
arccsc
x
a
+
a
x
2
1
−
a
2
x
2
+
C
{\displaystyle \int x\operatorname {arccsc} {\frac {x}{a}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}
∫
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)}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arccot
x
a
d
x
=
x
arccot
x
a
+
a
2
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} {\frac {x}{a}}\ dx=x\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}
∫
x
arccot
x
a
d
x
=
a
2
+
x
2
2
arccot
x
a
+
a
x
2
+
C
{\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}+C}
∫
x
2
arccot
x
a
d
x
=
x
3
3
arccot
x
a
+
a
x
2
6
−
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}
∫
x
n
arccot
x
a
d
x
=
x
n
+
1
n
+
1
arccot
x
a
+
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
∫
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)}
∫
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}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arctan
(
x
a
)
d
x
=
x
arctan
(
x
a
)
−
a
2
ln
(
1
+
x
2
a
2
)
+
C
{\displaystyle \int \arctan {\big (}{\frac {x}{a}}{\big )}dx=x\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}
∫
x
arctan
(
x
a
)
d
x
=
(
a
2
+
x
2
)
arctan
(
x
a
)
−
a
x
2
+
C
{\displaystyle \int x\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {(a^{2}+x^{2})\arctan {\big (}{\frac {x}{a}}{\big )}-ax}{2}}+C}
∫
x
2
arctan
(
x
a
)
d
x
=
x
3
3
arctan
(
x
a
)
−
a
x
2
6
+
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}
∫
x
n
arctan
(
x
a
)
d
x
=
x
n
+
1
n
+
1
arctan
(
x
a
)
−
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
|
x
+
x
x
2
−
1
x
2
|
+
C
{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫
arccsc
x
a
d
x
=
x
arccsc
x
a
+
a
ln
(
x
a
(
1
−
a
2
x
2
+
1
)
)
+
C
{\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ dx=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {({\frac {x}{a}}({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1))}+C}
∫
x
arccsc
x
a
d
x
=
x
2
2
arccsc
x
a
+
a
x
2
1
−
a
2
x
2
+
C
{\displaystyle \int x\operatorname {arccsc} {\frac {x}{a}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}
∫
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)}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arccot
x
a
d
x
=
x
arccot
x
a
+
a
2
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} {\frac {x}{a}}\ dx=x\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}
∫
x
arccot
x
a
d
x
=
a
2
+
x
2
2
arccot
x
a
+
a
x
2
+
C
{\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}+C}
∫
x
2
arccot
x
a
d
x
=
x
3
3
arccot
x
a
+
a
x
2
6
−
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}
∫
x
n
arccot
x
a
d
x
=
x
n
+
1
n
+
1
arccot
x
a
+
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
The following is a list of integrals (antiderivative formulas) for integrands that contain inverse trigonometric functions (also known as "arc functions"). For a complete list of integral formulas, see lists of integrals .
C is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
Note: There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin−1 , asin , or, as is used on this page, arcsin .
∫
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)}
∫
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}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arctan
(
x
a
)
d
x
=
x
arctan
(
x
a
)
−
a
2
ln
(
1
+
x
2
a
2
)
+
C
{\displaystyle \int \arctan {\big (}{\frac {x}{a}}{\big )}dx=x\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}
∫
x
arctan
(
x
a
)
d
x
=
(
a
2
+
x
2
)
arctan
(
x
a
)
−
a
x
2
+
C
{\displaystyle \int x\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {(a^{2}+x^{2})\arctan {\big (}{\frac {x}{a}}{\big )}-ax}{2}}+C}
∫
x
2
arctan
(
x
a
)
d
x
=
x
3
3
arctan
(
x
a
)
−
a
x
2
6
+
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}
∫
x
n
arctan
(
x
a
)
d
x
=
x
n
+
1
n
+
1
arctan
(
x
a
)
−
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
|
x
+
x
x
2
−
1
x
2
|
+
C
{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫
arccsc
x
a
d
x
=
x
arccsc
x
a
+
a
ln
(
x
a
(
1
−
a
2
x
2
+
1
)
)
+
C
{\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ dx=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {({\frac {x}{a}}({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1))}+C}
∫
x
arccsc
x
a
d
x
=
x
2
2
arccsc
x
a
+
a
x
2
1
−
a
2
x
2
+
C
{\displaystyle \int x\operatorname {arccsc} {\frac {x}{a}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}
∫
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)}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
(
1
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})+C}
∫
arccot
x
a
d
x
=
x
arccot
x
a
+
a
2
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arccot} {\frac {x}{a}}\ dx=x\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}
∫
x
arccot
x
a
d
x
=
a
2
+
x
2
2
arccot
x
a
+
a
x
2
+
C
{\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}+C}
∫
x
2
arccot
x
a
d
x
=
x
3
3
arccot
x
a
+
a
x
2
6
−
a
3
6
ln
(
a
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}
∫
x
n
arccot
x
a
d
x
=
x
n
+
1
n
+
1
arccot
x
a
+
a
n
+
1
∫
x
n
+
1
a
2
+
x
2
d
x
,
n
≠
−
1
{\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}
![{\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/4dedc009e8b17e7b7c02bda68c6d6b99d6578618)
