Popis integrala arc funkcija
Izvor: Hrvatska internetska enciklopedija
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
- [math]\displaystyle{ \int \arcsin \frac{x}{c} \ dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2} + C }[/math]
- [math]\displaystyle{ \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} + C }[/math]
- [math]\displaystyle{ \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} + C }[/math]
- [math]\displaystyle{ \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) + C }[/math]
Arkus kosinus
- [math]\displaystyle{ \int \arccos \frac{x}{c} \ dx = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2} + C }[/math]
- [math]\displaystyle{ \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} + C }[/math]
- [math]\displaystyle{ \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} + C }[/math]
Arkus tangens
- [math]\displaystyle{ \int \operatorname{arctg}\big( \frac{x}{c}\big) dx = x \operatorname{arctg} \big( \frac{x}{c} \big) - \frac{c}{2} \ln(c^2 + x^2) + C }[/math]
- [math]\displaystyle{ \int x \operatorname{arctg}\big( \frac{x}{c}\big) dx = \frac{ (c^2 + x^2) \operatorname{arctg} \big( \frac{x}{c} \big) - c x}{2} + C }[/math]
- [math]\displaystyle{ \int x^2 \operatorname{arctg}\big( \frac{x}{c}\big) dx = \frac{x^3}{3} \operatorname{arctg} \big(\frac{x}{c}\big) - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2} + C }[/math]
- [math]\displaystyle{ \int x^n \operatorname{arctg} \big( \frac{x}{c}\big) dx = \frac{x^{n + 1}}{n + 1} \operatorname{arctg} \big( \frac{x}{c} \big) - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx + C, \quad n \neq 1 }[/math]
Arkus sekans
- [math]\displaystyle{ \int \arcsec \frac{x}{c} \ dx = x \arcsec \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right| + C }[/math]
- [math]\displaystyle{ \int x \arcsec x \ dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right) + C }[/math]
- [math]\displaystyle{ \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) + C }[/math]
Arkus kotangens
- [math]\displaystyle{ \int \operatorname{arcctg} \frac{x}{c} \ dx = x \operatorname{arcctg} \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2) + C }[/math]
- [math]\displaystyle{ \int x \operatorname{arcctg} \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \operatorname{arcctg} \frac{x}{c} + \frac{c x}{2} + C }[/math]
- [math]\displaystyle{ \int x^2 \operatorname{arcctg} \frac{x}{c} \ dx = \frac{x^3}{3} \operatorname{arcctg} \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2) + C }[/math]
- [math]\displaystyle{ \int x^n \operatorname{arcctg} \frac{x}{c} \ dx = \frac{x^{n + 1}}{n+1} \operatorname{arcctg} \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx + C, \quad n \neq 1 }[/math]
Arkus kosekans
- [math]\displaystyle{ \int \arccsc \frac{x}{c} \ dx = x \arccsc \frac{x}{c} + {c} \ln{(\frac{x}{c}(\sqrt{1-\frac{c^2}{x^2}} + 1))} + C }[/math]
- [math]\displaystyle{ \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}} + C }[/math]
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