Lijst van integralen van inverse goniometrische functies

Dit artikel bevat een lijst van integralen van inverse goniometrische functies. De inverse functies van de goniometrische functies worden cyclometrische functies, arcfuncties of boogfuncties genoemd. Integralen zijn het onderwerp van studie van de integraalrekening.

Integralen met arcsinus

${\displaystyle \int \arcsin x\ \mathrm {d} x=x\arcsin x+{\sqrt {1-x^{2}\ }}}$

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

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

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

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

${\displaystyle \int \cos ^{n}x\arcsin x\ \mathrm {d} x=\left(x^{n^{2}+1}\arccos x+{\frac {x^{n}{\sqrt {1-x^{4}\ }}-nx^{n^{2}-1}\arccos x}{n^{2}-1}}+n\int x^{n^{2}-2}\arccos x\ \mathrm {d} x\right)}$

Integralen met arccosinus

${\displaystyle \int \arccos x\ \mathrm {d} x=x\arccos x-{\sqrt {1-x^{2}\ }}}$

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

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

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

Integralen met arctangens

${\displaystyle \int \arctan x\ \mathrm {d} x=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})}$

${\displaystyle \int \arctan \left({\frac {x}{a}}\right)\ \mathrm {d} x=x\arctan \left({\frac {x}{a}}\right)-{\frac {a}{2}}\ln \left(1+{\frac {x^{2}}{a^{2}}}\right)}$

${\displaystyle \int x\arctan \left({\frac {x}{a}}\right)\ \mathrm {d} x={\frac {(a^{2}+x^{2})\arctan \left({\frac {x}{a}}\right)-ax}{2}}}$

${\displaystyle \int x^{2}\arctan \left({\frac {x}{a}}\right)\ \mathrm {d} x={\frac {x^{3}}{3}}\arctan \left({\frac {x}{a}}\right)-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})}$

${\displaystyle \int x^{n}\arctan \left({\frac {x}{a}}\right)\ \mathrm {d} x={\frac {x^{n+1}}{n+1}}\arctan \left({\frac {x}{a}}\right)-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ \mathrm {d} x,\quad n\neq -1}$

Integralen met arccotangens

${\displaystyle \int \operatorname {arccot} x\ \mathrm {d} x=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})}$

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

${\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ \mathrm {d} x={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}}$

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

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

Integralen met arcsecans

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

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

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

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

Integralen met arccosecans

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

${\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ \mathrm {d} x=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {\left[\ {\frac {x}{a}}\left({\sqrt {1-{\frac {a^{2}}{x^{2}}}\ }}+1\right)\right]}}$

${\displaystyle \int x\operatorname {arccsc} {\frac {x}{a}}\ \mathrm {d} x={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}\ }}}$