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.
![{\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/8fb003b1d6c09a358355339c73e9599620f7de8e)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/796f0e12cce906ba160708a3671b20534bd34646)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/045a06bc40e3f77c8a95ea4f6de1b9a9b7c5aee8)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/136a51a8f7112f033b7bcd8806859614620d7800)
![{\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/556919284ee4e66a4a631855d0f606c7e1a9d0b1)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/066fd541b4b26c114408dada1fc15039b30857b0)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d46974fa874fa54ad8d1d8abdb01daff2cedccb2)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/e4357fa623db418807a1b0c051a9ca124c7c9a0e)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/50d3cba84fd4fa3a56ec8a6e13f54d8bc1c10145)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/cfc719939c18d9e173a6692de031e2274a5be53e)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d48cfa6dfb8c320aa614912627de4b5cce90c798)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/be44a5355f252c20f376b3387cc8ce25f704caea)
![{\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d7b36ac9151db90f64be275f4bad3bd133fb1c43)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/607cf9141ed99d1f3c1bb7f6000c47face4702ca)
![{\displaystyle \int \operatorname {arcctg} {\frac {x}{c}}\ dx=x\operatorname {arcctg} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/411d78fe45aede165ea83cd6392de86773c68047)
![{\displaystyle \int x\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {cx}{2}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4084f18d7e65928528c4e4f7f88bcffeb4067724)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/32e272076479970aa3eee626bdb1361de9cdf6c6)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/f0b94a86730a4462f420679b8c33ed5bfa3cd0f2)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/c8056a536c9aec0e07c6e2d78b54d5c857d7ff23)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d83809df1b218ffbfa4d1aeff2874a39ad247285)
Popis integrala inverznih triginometrijskih funkcija
[uredi | uredi kôd]
Koriste se supstitucija ili drugi oblici algebarskih manipulacija kako bi se dosegli integrali izlistani u tablici.