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

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

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

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

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

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

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

Integralen met arccosinus

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

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

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

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

Integralen met arctangens

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

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

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

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

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

Integralen met arccotangens

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

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

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

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

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

Integralen met arcsecans

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

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

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

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

Integralen met arccosecans

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

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

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

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Lijst van integralen

rationaal · irrationaal · exponentieel · logaritmisch · goniometrisch · invers goniometrisch · hyperbolisch · invers hyperbolisch