Seznam integrálů inverzních trigonometrických funkcí

Seznamy integrálů
Tabulka integrálů elementárních funkcí
racionální funkce iracionální funkce exponenciální funkce logaritmické funkce trigonometrické funkce inverzní trigonometrické funkce hyperbolické funkce inverzní hyperbolické funkce

Toto je seznam integrálů (primitivních funkcí) pro integrandy obsahující inverzní trigonometrické funkce.

Arkussinus

$\int \arcsin {\frac {x}{c}}\ \mathrm {d} x=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}$

$\int x\arcsin {\frac {x}{c}}\ \mathrm {d} x=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{2}\arcsin {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{n}\arcsin x\ \mathrm {d} x={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n-1}}+n\int x^{n-2}\arcsin x\ \mathrm {d} x\right)$

Arkuskosinus

$\int \arccos {\frac {x}{c}}\ \mathrm {d} x=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}$

$\int x\arccos {\frac {x}{c}}\ \mathrm {d} x=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arccos {\frac {x}{c}}-{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{2}\arccos {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{3}}{3}}\arccos {\frac {x}{c}}-{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}$

Arkustangens

$\int \arctan {\frac {x}{c}}\ \mathrm {d} x=x\arctan {\frac {x}{c}}-{\frac {c}{2}}\ln(c^{2}+x^{2})$

$\int x\arctan {\frac {x}{c}}\ \mathrm {d} x={\frac {(c^{2}+x^{2})\arctan {\frac {x}{c}}-cx}{2}}$

$\int x^{2}\arctan {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{3}}{3}}\arctan {\frac {x}{c}}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})$

$\int x^{n}\arctan {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{n+1}}{n+1}}\arctan {\frac {x}{c}}-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ \mathrm {d} x,\quad n\neq 1$

Arkussekans

$\int \operatorname {arcsec} {\frac {x}{c}}\ \mathrm {d} x=x\operatorname {arcsec} {\frac {x}{c}}+{\frac {x}{c|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|$

$\int x\operatorname {arcsec} x\ \mathrm {d} x={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)$

$\int x^{n}\operatorname {arcsec} x\ \mathrm {d} x={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+(1-n)\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\ \mathrm {d} x\right)\right]\right)$

Arkuskotangens

$\int \operatorname {arccot} {\frac {x}{c}}\ \mathrm {d} x=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})$

$\int x\operatorname {arccot} {\frac {x}{c}}\ \mathrm {d} x={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}$

$\int x^{2}\operatorname {arccot} {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})$

$\int x^{n}\operatorname {arccot} {\frac {x}{c}}\ \mathrm {d} x={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ \mathrm {d} x,\quad n\neq 1$