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Slijedi spisak integrala (antiderivacija funkcija) racionalnih funkcija za integrande koji sadrže inverzne trigonometrijske funkcije (poznate i kao “arc funkcije”). Za potpuni spisak integrala funkcija, pogledajte tabela integrala i spisak integrala.
Za rješavanje ovih integrala koriste se metoda supstitucije ili drugi oblici algebarskih manipulacija kako bi se dosegli integrali izlistani u tablici.
Napomena: Postoje tri uobičajene notacije za inverzne trigonometrijske funkcije. Arkus sinus funkcija bi se, na primijer, 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}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1f74b1d27103ee5523964842d6a0edecc3387725)
![{\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}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/7e2e7dd5fd424396705c93e9984fd37ab52f0e8b)
![{\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}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6fc49b205dd8a5a3f02ec647c4601f27dc570b36)
![{\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)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/da23343805dfc989be3082e69a9a68e91e3aaf9b)
![{\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/f10f314d5fb59856b4210288556a93ef2671a4c8)
![{\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}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4089b80626baef34255b4a9a1e3da35fc93f7bd6)
![{\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}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/87db748ddee03baeafca86b02fd354df6cf1047f)
![{\displaystyle \int \arctan \left({\frac {x}{c}}\right)dx=x\arctan \left({\frac {x}{c}}\right)-{\frac {c}{2}}\ln(c^{2}+x^{2})}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/7862f40bc0427cea57a5d5da8ba1defb9c31f2cd)
![{\displaystyle \int x\arctan \left({\frac {x}{c}}\right)dx={\frac {(c^{2}+x^{2})\arctan {\big (}{\frac {x}{c}}{\big )}-cx}{2}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/25b2ed107f8a70107ae445b91414add77f1255ab)
![{\displaystyle \int x^{2}\arctan \left({\frac {x}{c}}\right)dx={\frac {x^{3}}{3}}\arctan \left({\frac {x}{c}}\right)-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/f723bfad4d99c4a6c8e3a0f3a0369e2275579fda)
![{\displaystyle \int x^{n}\arctan \left({\frac {x}{c}}\right)dx={\frac {x^{n+1}}{n+1}}\arctan \left({\frac {x}{c}}\right)-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d15adb647fcbb227c493d9d1626e8a6efda0e3c8)
![{\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|}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/b5cc21718bb7d79436c73cd3664424a333008e14)
![{\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1794ec478f8766877e2d1fa636ca36d72c7821f7)
![{\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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4dedc009e8b17e7b7c02bda68c6d6b99d6578618)
![{\displaystyle \int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/8d0de77c2915221bad8f3bbbf2d618f4d8fc69b8)
![{\displaystyle \int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/de3c131126fc075cb53aeb5a20563186210a98ca)
![{\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})}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6e61fd76e59f54848909e95031025101fc4bdf6d)
![{\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}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/7ca0d0424569f9948919d79822ccb8fce27fed04)
![{\displaystyle \int \operatorname {arccsc} {\frac {x}{c}}\ dx=x\operatorname {arccsc} {\frac {x}{c}}+{c}\ln \left({\frac {x}{c}}\left({\sqrt {1-{\frac {c^{2}}{x^{2}}}}}+1\right)\right)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1aebbe1cf2a80bd9ec76eec57cbc368af7fb1275)
![{\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}}}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/32075f3cf2ea4f612854ced087065e1f91733d73)