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

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

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

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

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

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

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

Integralen met arccosinus

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

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

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

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

Integralen met arctangens

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

$\int \arctan\big( \frac{x}{a}\big) dx = x \arctan \big( \frac{x}{a} \big) - \frac{a}{2} \ln(1 + \frac{x^2}{a^2}) + C$

$\int x \arctan\big( \frac{x}{a}\big) dx = \frac{ (a^2 + x^2) \arctan \big( \frac{x}{a} \big) - a x}{2} + C$

$\int x^2 \arctan\big( \frac{x}{a}\big) dx = \frac{x^3}{3} \arctan \big(\frac{x}{a}\big) - \frac{a x^2}{6} + \frac{a^3}{6} \ln({a^2 + x^2}) + C$

$\int x^n \arctan \big( \frac{x}{a}\big) dx = \frac{x^{n + 1}}{n + 1} \arctan \big( \frac{x}{a} \big) - \frac{a}{n + 1} \int \frac{x^{n + 1}}{a^2 + x^2} \ dx, \quad n \neq -1$

Integralen met arccotangens

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

$\int \arccot \frac{x}{a} \ dx = x \arccot \frac{x}{a} + \frac{a}{2} \ln(a^2 + x^2) + C$

$\int x \arccot \frac{x}{a} \ dx = \frac{a^2 + x^2}{2} \arccot \frac{x}{a} + \frac{a x}{2} + C$

$\int x^2 \arccot \frac{x}{a} \ dx = \frac{x^3}{3} \arccot \frac{x}{a} + \frac{a x^2}{6} - \frac{a^3}{6} \ln(a^2 + x^2) + C$

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

Integralen met arcsecans

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

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

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

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

Integralen met arccosecans

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

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

$\int x \arccsc \frac{x}{a} \ dx = \frac{x^2}{2} \arccsc \frac{x}{a} + \frac{ax}{2} \sqrt{1-\frac{a^2}{x^2}} + C$