역삼각함수 적분표

아래 목록은 역삼각함수의 부정적분이다.

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

${\displaystyle \int \arcsin(ax)\,dx=x\arcsin(ax)+{\frac {\sqrt {1-a^{2}x^{2}{a}+C}$

${\displaystyle \int x\arcsin(ax)\,dx={\frac {x^{2}\arcsin(ax)}{2}-{\frac {\arcsin(ax)}{4\,a^{2}+{\frac {x{\sqrt {1-a^{2}x^{2}{4\,a}+C}$

${\displaystyle \int x^{2}\arcsin(ax)\,dx={\frac {x^{3}\arcsin(ax)}{3}+{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}{9\,a^{3}+C}$

${\displaystyle \int x^{m}\arcsin(ax)\,dx={\frac {x^{m+1}\arcsin(ax)}{m+1}\,-\,{\frac {a}{m+1}\int {\frac {x^{m+1}{\sqrt {1-a^{2}x^{2}\,dx\quad (m\neq -1)}$

${\displaystyle \int \arcsin(ax)^{2}\,dx=-2x+x\arcsin(ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}\arcsin(ax)}{a}+C}$

${\displaystyle \int \arcsin(ax)^{n}\,dx=x\arcsin(ax)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}x^{2}\arcsin(ax)^{n-1}{a}\,-\,n\,(n-1)\int \arcsin(ax)^{n-2}\,dx}$

${\displaystyle \int \arcsin(ax)^{n}\,dx={\frac {x\arcsin(ax)^{n+2}{(n+1)\,(n+2)}\,+\,{\frac {\sqrt {1-a^{2}x^{2}\arcsin(ax)^{n+1}{a\,(n+1)}\,-\,{\frac {1}{(n+1)\,(n+2)}\int \arcsin(ax)^{n+2}\,dx\quad (n\neq -1,-2)}$

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

${\displaystyle \int \arccos(ax)\,dx=x\arccos(ax)-{\frac {\sqrt {1-a^{2}x^{2}{a}+C}$

${\displaystyle \int x\arccos(ax)\,dx={\frac {x^{2}\arccos(ax)}{2}-{\frac {\arccos(ax)}{4\,a^{2}-{\frac {x{\sqrt {1-a^{2}x^{2}{4\,a}+C}$

${\displaystyle \int x^{2}\arccos(ax)\,dx={\frac {x^{3}\arccos(ax)}{3}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}{9\,a^{3}+C}$

${\displaystyle \int x^{m}\arccos(ax)\,dx={\frac {x^{m+1}\arccos(ax)}{m+1}\,+\,{\frac {a}{m+1}\int {\frac {x^{m+1}{\sqrt {1-a^{2}x^{2}\,dx\quad (m\neq -1)}$

${\displaystyle \int \arccos(ax)^{2}\,dx=-2x+x\arccos(ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}\arccos(ax)}{a}+C}$

${\displaystyle \int \arccos(ax)^{n}\,dx=x\arccos(ax)^{n}\,-\,{\frac {n{\sqrt {1-a^{2}x^{2}\arccos(ax)^{n-1}{a}\,-\,n\,(n-1)\int \arccos(ax)^{n-2}\,dx}$

${\displaystyle \int \arccos(ax)^{n}\,dx={\frac {x\arccos(ax)^{n+2}{(n+1)\,(n+2)}\,-\,{\frac {\sqrt {1-a^{2}x^{2}\arccos(ax)^{n+1}{a\,(n+1)}\,-\,{\frac {1}{(n+1)\,(n+2)}\int \arccos(ax)^{n+2}\,dx\quad (n\neq -1,-2)}$

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

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

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

${\displaystyle \int x^{2}\arctan(ax)\,dx={\frac {x^{3}\arctan(ax)}{3}+{\frac {\ln \left(a^{2}x^{2}+1\right)}{6\,a^{3}-{\frac {x^{2}{6\,a}+C}$

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

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

${\displaystyle \int \operatorname {arccot}(ax)\,dx=x\operatorname {arccot}(ax)+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}+C}$

${\displaystyle \int x\operatorname {arccot}(ax)\,dx={\frac {x^{2}\operatorname {arccot}(ax)}{2}+{\frac {\operatorname {arccot}(ax)}{2\,a^{2}+{\frac {x}{2\,a}+C}$

${\displaystyle \int x^{2}\operatorname {arccot}(ax)\,dx={\frac {x^{3}\operatorname {arccot}(ax)}{3}-{\frac {\ln \left(a^{2}x^{2}+1\right)}{6\,a^{3}+{\frac {x^{2}{6\,a}+C}$

${\displaystyle \int x^{m}\operatorname {arccot}(ax)\,dx={\frac {x^{m+1}\operatorname {arccot}(ax)}{m+1}+{\frac {a}{m+1}\int {\frac {x^{m+1}{a^{2}x^{2}+1}\,dx\quad (m\neq -1)}$

${\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C}$

${\displaystyle \int \operatorname {arcsec}(ax)\,dx=x\operatorname {arcsec}(ax)-{\frac {1}{a}\,\operatorname {arcosh} |ax|+C}$

${\displaystyle \int x\operatorname {arcsec}(ax)\,dx={\frac {x^{2}\operatorname {arcsec}(ax)}{2}-{\frac {x}{2\,a}{\sqrt {1-{\frac {1}{a^{2}x^{2}+C}$

${\displaystyle \int x^{2}\operatorname {arcsec}(ax)\,dx={\frac {x^{3}\operatorname {arcsec}(ax)}{3}\,-\,{\frac {\operatorname {arcosh} |ax|}{6\,a^{3}\,-\,{\frac {x^{2}{6\,a}{\sqrt {1-{\frac {1}{a^{2}x^{2}\,+\,C}$

${\displaystyle \int x^{m}\operatorname {arcsec}(ax)\,dx={\frac {x^{m+1}\operatorname {arcsec}(ax)}{m+1}\,-\,{\frac {1}{a\,(m+1)}\int {\frac {x^{m-1}{\sqrt {1-{\frac {1}{a^{2}x^{2}\,dx\quad (m\neq -1)}$

${\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}\right)\,+\,C=x\operatorname {arccsc}(x)\,+\,\operatorname {arcosh} |x|\,+\,C}$

${\displaystyle \int \operatorname {arccsc}(ax)\,dx=x\operatorname {arccsc}(ax)+{\frac {1}{a}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}+C}$

${\displaystyle \int x\operatorname {arccsc}(ax)\,dx={\frac {x^{2}\operatorname {arccsc}(ax)}{2}+{\frac {x}{2\,a}{\sqrt {1-{\frac {1}{a^{2}x^{2}+C}$

${\displaystyle \int x^{2}\operatorname {arccsc}(ax)\,dx={\frac {x^{3}\operatorname {arccsc}(ax)}{3}\,+\,{\frac {1}{6\,a^{3}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}\,+\,{\frac {x^{2}{6\,a}{\sqrt {1-{\frac {1}{a^{2}x^{2}\,+\,C}$

${\displaystyle \int x^{m}\operatorname {arccsc}(ax)\,dx={\frac {x^{m+1}\operatorname {arccsc}(ax)}{m+1}\,+\,{\frac {1}{a\,(m+1)}\int {\frac {x^{m-1}{\sqrt {1-{\frac {1}{a^{2}x^{2}\,dx\quad (m\neq -1)}$

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