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

∫ arcsin ⁡ x c   d x = x arcsin ⁡ x c + c 2 − x 2 {\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}}
∫ x arcsin ⁡ x c   d x = ( x 2 2 − c 2 4 ) arcsin ⁡ x c + x 4 c 2 − x 2 {\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}}}}
∫ x 2 arcsin ⁡ x c   d x = x 3 3 arcsin ⁡ x c + x 2 + 2 c 2 9 c 2 − x 2 {\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}}}}
∫ x n arcsin ⁡ x   d x = 1 n + 1 ( x n + 1 arcsin ⁡ x + x n 1 − x 2 − n x n − 1 arcsin ⁡ x n − 1 + n ∫ x n − 2 arcsin ⁡ x   d x ) {\displaystyle \int x^{n}\arcsin x\ dx={\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\ dx\right)}

Arkus kosinus

∫ arccos ⁡ x c   d x = x arccos ⁡ x c − c 2 − x 2 {\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}
∫ x arccos ⁡ x c   d x = ( x 2 2 − c 2 4 ) arccos ⁡ x c − x 4 c 2 − x 2 {\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}}}}
∫ x 2 arccos ⁡ x c   d x = x 3 3 arccos ⁡ x c − x 2 + 2 c 2 9 c 2 − x 2 {\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}}}}

Arkus tangens

∫ arctan ⁡ ( x c ) d x = x arctan ⁡ ( x c ) − c 2 ln ⁡ ( c 2 + x 2 ) {\displaystyle \int \arctan {\big (}{\frac {x}{c}}{\big )}dx=x\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫ x arctan ⁡ ( x c ) d x = ( c 2 + x 2 ) arctan ⁡ ( x c ) − c x 2 {\displaystyle \int x\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {(c^{2}+x^{2})\arctan {\big (}{\frac {x}{c}}{\big )}-cx}{2}}}
∫ x 2 arctan ⁡ ( x c ) d x = x 3 3 arctan ⁡ ( x c ) − c x 2 6 + c 3 6 ln ⁡ c 2 + x 2 {\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{c}}{\big )}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}}
∫ x n arctan ⁡ ( x c ) d x = x n + 1 n + 1 arctan ⁡ ( x c ) − c n + 1 ∫ x n + 1 c 2 + x 2   d x , n ≠ 1 {\displaystyle \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

∫ arcsec ⁡ x c   d x = x arcsec ⁡ x c + x c | x | ln ⁡ | x ± x 2 − 1 | {\displaystyle \int \operatorname {arcsec} {\frac {x}{c}}\ dx=x\operatorname {arcsec} {\frac {x}{c}}+{\frac {x}{c|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|}
∫ x arcsec ⁡ x   d x = 1 2 ( x 2 arcsec ⁡ x − x 2 − 1 ) {\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)}
∫ x n arcsec ⁡ x   d x = 1 n + 1 ( x n + 1 arcsec ⁡ x − 1 n [ x n − 1 x 2 − 1 + ( 1 − n ) ( x n − 1 arcsec ⁡ x + ( 1 − n ) ∫ x n − 2 arcsec ⁡ x   d x ) ] ) {\displaystyle \int x^{n}\operatorname {arcsec} x\ dx={\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\ dx\right)\right]\right)}

Arkus kotangens

∫ arccot ⁡ x c   d x = x arccot ⁡ x c + c 2 ln ⁡ ( c 2 + x 2 ) {\displaystyle \int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}
∫ x arccot ⁡ x c   d x = c 2 + x 2 2 arccot ⁡ x c + c x 2 {\displaystyle \int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}}
∫ x 2 arccot ⁡ x c   d x = x 3 3 arccot ⁡ x c + c x 2 6 − c 3 6 ln ⁡ ( c 2 + x 2 ) {\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})}
∫ x n arccot ⁡ x c   d x = x n + 1 n + 1 arccot ⁡ x c + c n + 1 ∫ x n + 1 c 2 + x 2   d x , n ≠ 1 {\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1}

Arkus kosekans

∫ arccsc ⁡ x c   d x = x arccsc ⁡ x c + c ln ⁡ ( x c ( 1 − c 2 x 2 + 1 ) ) {\displaystyle \int \operatorname {arccsc} {\frac {x}{c}}\ dx=x\operatorname {arccsc} {\frac {x}{c}}+{c}\ln {({\frac {x}{c}}({\sqrt {1-{\frac {c^{2}}{x^{2}}}}}+1))}}
∫ x arccsc ⁡ x c   d x = x 2 2 arccsc ⁡ x c + c x 2 1 − c 2 x 2 {\displaystyle \int x\operatorname {arccsc} {\frac {x}{c}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{c}}+{\frac {cx}{2}}{\sqrt {1-{\frac {c^{2}}{x^{2}}}}}}