역삼각함수 적분표

아래 목록은 역삼각함수의 부정적분이다.

## 아크사인

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

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

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

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

: \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^2x^2}}\,dx\quad(m\ne-1)

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

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

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

## 아크코사인

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

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

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

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

: \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^2x^2}}\,dx\quad(m\ne-1)

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

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

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

## 아크탄젠트

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

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

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

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

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

## 아크코탄젠트

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

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

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

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

: \int x^m\arccot(ax)\,dx=
 \frac{x^{m+1}\arccot(ax)}{m+1}+
 \frac{a}{m+1}\int \frac{x^{m+1}}{a^2x^2+1}\,dx\quad(m\ne-1)

## 아크시컨트

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

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

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

: \int x^m\arcsec(ax)\,dx=
 \frac{x^{m+1}\arcsec(ax)}{m+1}\,-\,
 \frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\quad(m\ne-1)

## 아크코시컨트

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

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

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

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

: \int x^m\arccsc(ax)\,dx=
 \frac{x^{m+1}\arccsc(ax)}{m+1}\,+\,
 \frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\quad(m\ne-1)

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