Arkus sinus
uredi
∫ arcsin x c d x = x arcsin x c + c 2 − x 2 + C {\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}+C} ∫ x arcsin x c d x = ( x 2 2 − c 2 4 ) arcsin x c + x 4 c 2 − x 2 + C {\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} ∫ x 2 arcsin x c d x = x 3 3 arcsin x c + x 2 + 2 c 2 9 c 2 − x 2 + C {\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} ∫ 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 ) + C {\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)+C} Arkus kosinus
uredi
∫ arccos x c d x = x arccos x c − c 2 − x 2 + C {\displaystyle \int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}+C} ∫ x arccos x c d x = ( x 2 2 − c 2 4 ) arccos x c − x 4 c 2 − x 2 + C {\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} ∫ x 2 arccos x c d x = x 3 3 arccos x c − x 2 + 2 c 2 9 c 2 − x 2 + C {\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} Arkus tangens
uredi
∫ arctg ( x c ) d x = x arctg ( x c ) − c 2 ln ( c 2 + x 2 ) + C {\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} ∫ x arctg ( x c ) d x = ( c 2 + x 2 ) arctg ( x c ) − c x 2 + C {\displaystyle \int x\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx={\frac {(c^{2}+x^{2})\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-cx}{2}}+C} ∫ x 2 arctg ( x c ) d x = x 3 3 arctg ( x c ) − c x 2 6 + c 3 6 ln c 2 + x 2 + C {\displaystyle \int x^{2}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}dx={\frac {x^{3}}{3}}\operatorname {arctg} {\big (}{\frac {x}{c}}{\big )}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}+C} ∫ x n arctg ( x c ) d x = x n + 1 n + 1 arctg ( x c ) − c n + 1 ∫ x n + 1 c 2 + x 2 d x + C , n ≠ 1 {\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} Arkus sekans
uredi
∫ arcsec x c d x = x arcsec x c + x c | x | ln | x ± x 2 − 1 | + C {\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|+C} ∫ x arcsec x d x = 1 2 ( x 2 arcsec x − x 2 − 1 ) + C {\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)+C} ∫ 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 ) ] ) + C {\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)+C} Arkus kotangens
uredi
∫ arcctg x c d x = x arcctg x c + c 2 ln ( c 2 + x 2 ) + C {\displaystyle \int \operatorname {arcctg} {\frac {x}{c}}\ dx=x\operatorname {arcctg} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})+C} ∫ x arcctg x c d x = c 2 + x 2 2 arcctg x c + c x 2 + C {\displaystyle \int x\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {cx}{2}}+C} ∫ x 2 arcctg x c d x = x 3 3 arcctg x c + c x 2 6 − c 3 6 ln ( c 2 + x 2 ) + C {\displaystyle \int x^{2}\operatorname {arcctg} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arcctg} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})+C} ∫ x n arcctg x c d x = x n + 1 n + 1 arcctg x c + c n + 1 ∫ x n + 1 c 2 + x 2 d x + C , n ≠ 1 {\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} Arkus kosekans
uredi
∫ arccsc x c d x = x arccsc x c + c ln ( x c ( 1 − c 2 x 2 + 1 ) ) + C {\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))}+C} ∫ x arccsc x c d x = x 2 2 arccsc x c + c x 2 1 − c 2 x 2 + C {\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}}}}}+C} Popis integrala inverznih triginometrijskih funkcija
uredi
Koriste se supstitucija ili drugi oblici algebarskih manipulacija kako bi se dosegli integrali izlistani u tablici.
∫ arcsin x d x = x arcsin x + 1 − x 2 + C {\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}
∫ arccos x d x = x arccos x − 1 − x 2 + C {\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}
∫ arctg x d x = x arctg x − 1 2 ln | 1 + x 2 | + C {\displaystyle \int \operatorname {arctg} x\,dx=x\operatorname {arctg} x-{\frac {1}{2}}\ln |1+x^{2}|+C}
∫ arccsc x d x = x arccsc x + ln | x + x x 2 − 1 x 2 | + C {\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫ arcsec x d x = x arcsec x − ln | x + x x 2 − 1 x 2 | + C {\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}
∫ arcctg x d x = x arcctg x + 1 2 ln | 1 + x 2 | + C {\displaystyle \int \operatorname {arcctg} x\,dx=x\operatorname {arcctg} x+{\frac {1}{2}}\ln |1+x^{2}|+C}