Tot seguit es presenta una llista de primitives d'inverses de funcions trigonomètriques. Per consultar una llista completa de primitives de tota mena de funcions adreceu-vos a taula d'integrals
La constant c se suposa diferent de zero.
Nota: Hi ha tres notacions habituals per a referir-se a les inverses de les funcions trigonomètriques. La inversa de la funció sSinus, per exemple, es pot escriure com sin−1, asin, o, tal com es fa en aquest article, arcsin.
Arcsinus
![{\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c37a9ab1c1e81a2fb12fc9fb61e1b96caa966a3)
![{\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}}](https://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/da23343805dfc989be3082e69a9a68e91e3aaf9b)
Arccosinus
![{\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4156a995781a73084bc0d85bfa09ebc40b01268b)
![{\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}](https://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/87db748ddee03baeafca86b02fd354df6cf1047f)
Arctangent
![{\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln |1+x^{2}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d3e1db9dec5aa0a214faea305b0af14e82a38b)
![{\displaystyle \int \arctan {\big (}{\frac {x}{c}}{\big )}dx=x\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{2}}\ln(1+{\frac {x^{2}}{c^{2}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82abea1c514cd0b3dfc43d79d013f00d1640a82)
![{\displaystyle \int x\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {(c^{2}+x^{2})\arctan {\big (}{\frac {x}{c}}{\big )}-cx}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c01ef8e21b2fed29463a9ac313a978af62e594d)
![{\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}}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4414a88fa4778f9d00e7f3894e9dc61f19619f5)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7fe953c0d95eb8365898a8ae02ece2619d0e2ad)
Arccosecant
![{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a619f60419145e6720f986284f126abc54bbeee)
![{\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))}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb96f30f62582afb6d4dcebc7853eafd3c86124)
![{\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://wikimedia.org/api/rest_v1/media/math/render/svg/32075f3cf2ea4f612854ced087065e1f91733d73)
Arcsecant
![{\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23806ed3b858880b48199a5dd1581698a37332ea)
![{\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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/4dedc009e8b17e7b7c02bda68c6d6b99d6578618)
Arccotangent
![{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln |1+x^{2}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0148a555a211ba90748faea6749b7f2a54f89ebb)
![{\displaystyle \int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}](https://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/7ca0d0424569f9948919d79822ccb8fce27fed04)
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Funcions trigonomètriques | | |
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Funcions trigonomètriques inverses | |
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