다음은 역 삼각 함수(inverse trigonometric function)를 포함하는 표현의 부정 적분(indefinite integral) (역도함수(antiderivative))의 목록입니다. 적분 공식의 전체 목록은 적분의 목록(lists of integrals)을 참조하십시오.
- 역 삼각 함수는 역시 "원호 함수"로 알려져 있습니다.
- C는 임의적인 적분의 상수(constant of integration)에 사용되며, 그것은 만약 어떤 점에서 적분의 값에 대한 무언가가 알려진 있으면 오직 결정될 수 있습니다. 따라서 각 함수는 무한한 숫자의 역도함수를 가집니다.
- 역 삼각 함수에 대해 세 가지 공통적인 표기법이 있습니다. 아크사인 함수는, 예를 들어, \(\sin^{-1}\), asin 또는 이 페이지에서 사용된 것처럼 arcsin으로 쓰일 수 있습니다.
- 아래의 각 역 삼각 적분화 공식에 대해, 역 쌍곡 함수의 적분의 목록(list of integrals of inverse hyperbolic functions)에서 해당 공식이 있습니다.
Arcsine function integration formulas
\(\displaystyle \int\arcsin(x)\,dx=
x\arcsin(x)+
{\sqrt{1-x^2}}+C\)
\(\displaystyle \int\arcsin(ax)\,dx=
x\arcsin(ax)+
\frac{\sqrt{1-a^2x^2}}{a}+C\)
\(\displaystyle \int x\arcsin(ax)\,dx=
\frac{x^2\arcsin(ax)}{2}-
\frac{\arcsin(ax)}{4\,a^2}+
\frac{x\sqrt{1-a^2x^2}}{4\,a}+C\)
\(\displaystyle \int x^2\arcsin(ax)\,dx=
\frac{x^3\arcsin(ax)}{3}+
\frac{\left(a^2x^2+2\right)\sqrt{1-a^2x^2}}{9\,a^3}+C\)
\(\displaystyle \int x^m\arcsin(ax)\,dx=
\frac{x^{m+1}\arcsin(ax)}{m+1}\,-\,
\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2x^2}}\,dx\quad(m\ne-1)\)
\(\displaystyle \int\arcsin(ax)^2\,dx=
-2x+x\arcsin(ax)^2+
\frac{2\sqrt{1-a^2x^2}\arcsin(ax)}{a}+C\)
\(\displaystyle \int\arcsin(ax)^n\,dx=
x\arcsin(ax)^n\,+\,
\frac{n\sqrt{1-a^2x^2}\arcsin(ax)^{n-1}}{a}\,-\,
n\,(n-1)\int\arcsin(ax)^{n-2}\,dx\)
\(\displaystyle \int\arcsin(ax)^n\,dx=
\frac{x\arcsin(ax)^{n+2}}{(n+1)\,(n+2)}\,+\,
\frac{\sqrt{1-a^2x^2}\arcsin(ax)^{n+1}}{a\,(n+1)}\,-\,
\frac{1}{(n+1)\,(n+2)}\int\arcsin(ax)^{n+2}\,dx\quad(n\ne-1,-2)\)
Arccosine function integration formulas
\(\displaystyle \int\arccos(x)\,dx=
x\arccos(x)-
{\sqrt{1-x^2}}+C\)
\(\displaystyle \int\arccos(ax)\,dx=
x\arccos(ax)-
\frac{\sqrt{1-a^2x^2}}{a}+C\)
\(\displaystyle \int x\arccos(ax)\,dx=
\frac{x^2\arccos(ax)}{2}-
\frac{\arccos(ax)}{4\,a^2}-
\frac{x\sqrt{1-a^2x^2}}{4\,a}+C\)
\(\displaystyle \int x^2\arccos(ax)\,dx=
\frac{x^3\arccos(ax)}{3}-
\frac{\left(a^2x^2+2\right)\sqrt{1-a^2x^2}}{9\,a^3}+C\)
\(\displaystyle \int x^m\arccos(ax)\,dx=
\frac{x^{m+1}\arccos(ax)}{m+1}\,+\,
\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2x^2}}\,dx\quad(m\ne-1)\)
\(\displaystyle \int\arccos(ax)^2\,dx=
-2x+x\arccos(ax)^2-
\frac{2\sqrt{1-a^2x^2}\arccos(ax)}{a}+C\)
\(\displaystyle \int\arccos(ax)^n\,dx=
x\arccos(ax)^n\,-\,
\frac{n\sqrt{1-a^2x^2}\arccos(ax)^{n-1}}{a}\,-\,
n\,(n-1)\int\arccos(ax)^{n-2}\,dx\)
\(\displaystyle \int\arccos(ax)^n\,dx=
\frac{x\arccos(ax)^{n+2}}{(n+1)\,(n+2)}\,-\,
\frac{\sqrt{1-a^2x^2}\arccos(ax)^{n+1}}{a\,(n+1)}\,-\,
\frac{1}{(n+1)\,(n+2)}\int\arccos(ax)^{n+2}\,dx\quad(n\ne-1,-2)\)
Arctangent function integration formulas
\(\displaystyle \int\arctan(x)\,dx=
x\arctan(x)-
\frac{\ln\left(x^2+1\right)}{2}+C\)
\(\displaystyle \int\arctan(ax)\,dx=
x\arctan(ax)-
\frac{\ln\left(a^2x^2+1\right)}{2\,a}+C\)
\(\displaystyle \int x\arctan(ax)\,dx=
\frac{x^2\arctan(ax)}{2}+
\frac{\arctan(ax)}{2\,a^2}-\frac{x}{2\,a}+C\)
\(\displaystyle \int x^2\arctan(ax)\,dx=
\frac{x^3\arctan(ax)}{3}+
\frac{\ln\left(a^2x^2+1\right)}{6\,a^3}-\frac{x^2}{6\,a}+C\)
\(\displaystyle \int x^m\arctan(ax)\,dx=
\frac{x^{m+1}\arctan(ax)}{m+1}-
\frac{a}{m+1}\int \frac{x^{m+1}}{a^2x^2+1}\,dx\quad(m\ne-1)\)
Arccotangent function integration formulas
\(\displaystyle \int\text{arccot}(x)\,dx=
x\text{arccot}(x)+
\frac{\ln\left(x^2+1\right)}{2}+C\)
\(\displaystyle \int\text{arccot}(ax)\,dx=
x\text{arccot}(ax)+
\frac{\ln\left(a^2x^2+1\right)}{2\,a}+C\)
\(\displaystyle \int x\text{arccot}(ax)\,dx=
\frac{x^2\text{arccot}(ax)}{2}+
\frac{\text{arccot}(ax)}{2\,a^2}+\frac{x}{2\,a}+C\)
\(\displaystyle \int x^2\text{arccot}(ax)\,dx=
\frac{x^3\text{arccot}(ax)}{3}-
\frac{\ln\left(a^2x^2+1\right)}{6\,a^3}+\frac{x^2}{6\,a}+C\)
\(\displaystyle \int x^m\text{arccot}(ax)\,dx=
\frac{x^{m+1}\text{arccot}(ax)}{m+1}+
\frac{a}{m+1}\int \frac{x^{m+1}}{a^2x^2+1}\,dx\quad(m\ne-1)\)
Arcsecant function integration formulas
\(\displaystyle \int\text{arcsec}(x)\,dx=
x\text{arcsec}(x)-\operatorname{arcosh}|x|+C\)
\(\displaystyle \int\text{arcsec}(ax)\,dx=
x\text{arcsec}(ax)-
\frac{1}{a}\,\operatorname{arcosh}|ax|+C\)
\(\displaystyle \int x\text{arcsec}(ax)\,dx=
\frac{x^2\text{arcsec}(ax)}{2}-
\frac{x}{2\,a}\sqrt{1-\frac{1}{a^2x^2}}+C\)
\(\displaystyle \int x^2\text{arcsec}(ax)\,dx=
\frac{x^3\text{arcsec}(ax)}{3}\,-\,
\frac{\operatorname{arcosh}|ax|}{6\,a^3}\,-\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2x^2}}\,+\,C\)
\(\displaystyle \int x^m\text{arcsec}(ax)\,dx=
\frac{x^{m+1}\text{arcsec}(ax)}{m+1}\,-\,
\frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\quad(m\ne-1)\)
Arccosecant function integration formulas
\(\displaystyle \int\text{arccsc}(x)\,dx=
x\text{arccsc}(x) \, + \,
\ln\left(\left|x\right|+\sqrt{x^2-1}\right)\,+\,C=
x\text{arccsc}(x)\,+\,
\operatorname{arcosh}|x|\,+\,C \)
\(\displaystyle \int\text{arccsc}(ax)\,dx=
x\text{arccsc}(ax)+
\frac{1}{a}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2x^2}}+C\)
\(\displaystyle \int x\text{arccsc}(ax)\,dx=
\frac{x^2\text{arccsc}(ax)}{2}+
\frac{x}{2\,a}\sqrt{1-\frac{1}{a^2x^2}}+C\)
\(\displaystyle \int x^2\text{arccsc}(ax)\,dx=
\frac{x^3\text{arccsc}(ax)}{3}\,+\,
\frac{1}{6\,a^3}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2x^2}}\,+\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2x^2}}\,+\,C\)
\(\displaystyle \int x^m\text{arccsc}(ax)\,dx=
\frac{x^{m+1}\text{arccsc}(ax)}{m+1}\,+\,
\frac{1}{a\,(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2x^2}}}\,dx\quad(m\ne-1)\)