Slijedi popis integrala (antiderivacija funkcija) racionalnih funkcija za integrande koji sadrže inverzne trigonometrijske funkcije (poznate i kao “arc funkcije”). Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala .
Bilješka: Postoje tri uobičajene notacije za inverzne trigonometrijske funkcije. Arkus sinus funkcija bi se primjerice mogla zapisati kao sin−1 , asin, ili kao što je korišteno u ovom članku, kao arcsin.
∫
arcsin
x
c
d
x
=
x
arcsin
x
c
+
c
2
−
x
2
+
C
{\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}+C}
∫
x
arcsin
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arcsin
x
c
+
x
4
c
2
−
x
2
+
C
{\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}}}+C}
∫
x
2
arcsin
x
c
d
x
=
x
3
3
arcsin
x
c
+
x
2
+
2
c
2
9
c
2
−
x
2
+
C
{\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}}}+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
)
+
C
{\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)+C}
∫
arccos
x
c
d
x
=
x
arccos
x
c
−
c
2
−
x
2
+
C
{\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}+C}
∫
x
arccos
x
c
d
x
=
(
x
2
2
−
c
2
4
)
arccos
x
c
−
x
4
c
2
−
x
2
+
C
{\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}}}+C}
∫
x
2
arccos
x
c
d
x
=
x
3
3
arccos
x
c
−
x
2
+
2
c
2
9
c
2
−
x
2
+
C
{\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}}}+C}
∫
arctg
(
x
c
)
d
x
=
x
arctg
(
x
c
)
−
c
2
ln
(
c
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx=x\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{2}}\ln(c^{2}+x^{2})+C}
∫
x
arctg
(
x
c
)
d
x
=
(
c
2
+
x
2
)
arctg
(
x
c
)
−
c
x
2
+
C
{\displaystyle \int x\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx={\frac {(c^{2}+x^{2})\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-cx}{2}}+C}
∫
x
2
arctg
(
x
c
)
d
x
=
x
3
3
arctg
(
x
c
)
−
c
x
2
6
+
c
3
6
ln
c
2
+
x
2
+
C
{\displaystyle \int x^{2}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{3}}{3}}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}+C}
∫
x
n
arctg
(
x
c
)
d
x
=
x
n
+
1
n
+
1
arctg
(
x
c
)
−
c
n
+
1
∫
x
n
+
1
c
2
+
x
2
d
x
+
C
,
n
≠
1
{\displaystyle \int x^{n}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{n+1}}{n+1}}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx+C,\quad n\neq 1}
∫
arcsec
x
c
d
x
=
x
arcsec
x
c
+
x
c
|
x
|
ln
|
x
±
x
2
−
1
|
+
C
{\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|+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
)
]
)
+
C
{\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)+C}
∫
arcctg
x
c
d
x
=
x
arcctg
x
c
+
c
2
ln
(
c
2
+
x
2
)
+
C
{\displaystyle \int \operatorname {arcctg} {\frac {x}{c}}\ dx=x\operatorname {arcctg} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})+C}
∫
x
arcctg
x
c
d
x
=
c
2
+
x
2
2
arcctg
x
c
+
c
x
2
+
C
{\displaystyle \int x\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {cx}{2}}+C}
∫
x
2
arcctg
x
c
d
x
=
x
3
3
arcctg
x
c
+
c
x
2
6
−
c
3
6
ln
(
c
2
+
x
2
)
+
C
{\displaystyle \int x^{2}\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})+C}
∫
x
n
arcctg
x
c
d
x
=
x
n
+
1
n
+
1
arcctg
x
c
+
c
n
+
1
∫
x
n
+
1
c
2
+
x
2
d
x
+
C
,
n
≠
1
{\displaystyle \int x^{n}\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx+C,\quad n\neq 1}
∫
arccsc
x
c
d
x
=
x
arccsc
x
c
+
c
ln
(
x
c
(
1
−
c
2
x
2
+
1
)
)
+
C
{\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))}+C}
∫
x
arccsc
x
c
d
x
=
x
2
2
arccsc
x
c
+
c
x
2
1
−
c
2
x
2
+
C
{\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}}}}}+C}
Popis integrala inverznih triginometrijskih funkcija [ uredi | uredi kôd ]
Koriste se supstitucija ili drugi oblici algebarskih manipulacija kako bi se dosegli integrali izlistani u tablici.
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
x
2
+
C
{\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}
∫
arccos
x
d
x
=
x
arccos
x
−
1
−
x
2
+
C
{\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-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}
∫
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}
∫
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}
∫
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}