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)