Arkus sinus
uredi
∫
arcsin
x
c
d
x
=
x
arcsin
x
c
+
c
2
−
x
2
{\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}}
∫
x
arcsin
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arcsin
x
c
+
x
4
c
2
−
x
2
{\displaystyle \int x\arcsin {\frac {x}{c}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}
∫
x
2
arcsin
x
c
d
x
=
x
3
3
arcsin
x
c
+
x
2
+
2
c
2
9
c
2
−
x
2
{\displaystyle \int x^{2}\arcsin {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}
∫
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)}
Arkus kosinus
uredi
∫
arccos
x
c
d
x
=
x
arccos
x
c
−
c
2
−
x
2
{\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}
∫
x
arccos
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arccos
x
c
−
x
4
c
2
−
x
2
{\displaystyle \int x\arccos {\frac {x}{c}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arccos {\frac {x}{c}}-{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}
∫
x
2
arccos
x
c
d
x
=
x
3
3
arccos
x
c
−
x
2
+
2
c
2
9
c
2
−
x
2
{\displaystyle \int x^{2}\arccos {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arccos {\frac {x}{c}}-{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}
Arkus tangens
uredi
∫
arctan
(
x
c
)
d
x
=
x
arctan
(
x
c
)
−
c
2
ln
(
c
2
+
x
2
)
{\displaystyle \int \arctan {\big (}{\frac {x}{c}}{\big )}dx=x\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫
x
arctan
(
x
c
)
d
x
=
(
c
2
+
x
2
)
arctan
(
x
c
)
−
c
x
2
{\displaystyle \int x\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {(c^{2}+x^{2})\arctan {\big (}{\frac {x}{c}}{\big )}-cx}{2}}}
∫
x
2
arctan
(
x
c
)
d
x
=
x
3
3
arctan
(
x
c
)
−
c
x
2
6
+
c
3
6
ln
c
2
+
x
2
{\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}}
∫
x
n
arctan
(
x
c
)
d
x
=
x
n
+
1
n
+
1
arctan
(
x
c
)
−
c
n
+
1
∫
x
n
+
1
c
2
+
x
2
d
x
,
n
≠
1
{\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1}
Arkus sekans
uredi
∫
arcsec
x
c
d
x
=
x
arcsec
x
c
+
x
c
|
x
|
ln
|
x
±
x
2
−
1
|
{\displaystyle \int \operatorname {arcsec} {\frac {x}{c}}\ dx=x\operatorname {arcsec} {\frac {x}{c}}+{\frac {x}{c|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|}
∫
x
arcsec
x
d
x
=
1
2
(
x
2
arcsec
x
−
x
2
−
1
)
{\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)}
∫
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)}
Arkus kotangens
uredi
∫
arccot
x
c
d
x
=
x
arccot
x
c
+
c
2
ln
(
c
2
+
x
2
)
{\displaystyle \int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫
x
arccot
x
c
d
x
=
c
2
+
x
2
2
arccot
x
c
+
c
x
2
{\displaystyle \int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}}
∫
x
2
arccot
x
c
d
x
=
x
3
3
arccot
x
c
+
c
x
2
6
−
c
3
6
ln
(
c
2
+
x
2
)
{\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})}
∫
x
n
arccot
x
c
d
x
=
x
n
+
1
n
+
1
arccot
x
c
+
c
n
+
1
∫
x
n
+
1
c
2
+
x
2
d
x
,
n
≠
1
{\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1}
Arkus kosekans
uredi
∫
arccsc
x
c
d
x
=
x
arccsc
x
c
+
c
ln
(
x
c
(
1
−
c
2
x
2
+
1
)
)
{\displaystyle \int \operatorname {arccsc} {\frac {x}{c}}\ dx=x\operatorname {arccsc} {\frac {x}{c}}+{c}\ln {({\frac {x}{c}}({\sqrt {1-{\frac {c^{2}}{x^{2}}}}}+1))}}
∫
x
arccsc
x
c
d
x
=
x
2
2
arccsc
x
c
+
c
x
2
1
−
c
2
x
2
{\displaystyle \int x\operatorname {arccsc} {\frac {x}{c}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{c}}+{\frac {cx}{2}}{\sqrt {1-{\frac {c^{2}}{x^{2}}}}}}
![{\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)
