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

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

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

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

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