Lijst van integralen van inverse goniometrische functies

Dit artikel bevat een lijst van integralen van inverse goniometrische functies. Deze functies worden ook wel cyclometrische of boogfuncties genoemd.

Integralen met arcsinus

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

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

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

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

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

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

Integralen met arccosinus

$\int \arccos x\,\mathrm {d} x=x\arccos x-{\sqrt {1-x^{2}}}+C$

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

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

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

Integralen met arctangens

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

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

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

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

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

Integralen met arccotangens

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

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

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

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

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

Integralen met arcsecans

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

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

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

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

Integralen met arccosecans

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

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

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