Popis integrala arc funkcija

Slijedi popis integrala (antiderivacija funkcija) racionalnih funkcija za integrande koji sadrže inverzne trigonometrijske funkcije (poznate i kao “arc funkcije”). Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala.

Bilješka: Postoje tri uobičajene notacije za inverzne trigonometrijske funkcije. Arkus sinus funkcija bi se primjerice mogla zapisati kao sin⁻¹, asin, ili kao što je korišteno u ovom članku, kao arcsin.

Arkus sinus

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

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

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

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

Arkus kosinus

$\int \arccos \frac{x}{c} \ dx = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}$

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

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

Arkus tangens

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

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

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

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

Arkus sekans

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

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

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

Arkus kotangens

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

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

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

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

Arkus kosekans

$\int \arccsc \frac{x}{c} \ dx = x \arccsc \frac{x}{c} + {c} \ln{(\frac{x}{c}(\sqrt{1-\frac{c^2}{x^2}} + 1))}$

$\int x \arccsc \frac{x}{c} \ dx = \frac{x^2}{2} \arccsc \frac{x}{c} + \frac{cx}{2} \sqrt{1-\frac{c^2}{x^2}}$