역쌍곡선함수 적분표

1. 개요

역쌍곡선 함수 적분표는 주요 역쌍곡선 함수들의 적분 공식들을 모아 놓은 문서이다. 역쌍곡사인 함수, 역쌍곡코사인 함수, 역쌍곡탄젠트 함수, 역쌍곡코탄젠트 함수, 역쌍곡시컨트 함수, 역쌍곡코시컨트 함수 등 다양한 역쌍곡선 함수와 관련된 적분 공식들을 제공하며, 각 공식에서 C는 적분 상수를 나타낸다.

2. 역쌍곡선 함수 적분 공식

다음은 주요 역쌍곡선 함수들의 적분 공식들이다. 여기서 C는 적분 상수를 나타낸다.

* 역쌍곡사인 함수 (arsinh)
* 역쌍곡코사인 함수 (arcosh) (en: arcosh)
* 역쌍곡탄젠트 함수 (artanh)
* 역쌍곡코탄젠트 함수 (arcoth)
* 역쌍곡시컨트 함수 (arsech)
* 역쌍곡코시컨트 함수 (arcsch)

2.1. 역쌍곡사인 함수 (arsinh)

다음은 역쌍곡사인 함수(arsinh)의 적분 공식들이다.

:$\backslash int\backslash operatorname\{arsinh\}(ax)\backslash ,dx=\; x\backslash operatorname\{arsinh\}(ax)-\backslash frac\{\backslash sqrt\{a^2x^2+1\}\}\{a\}+C$

:$\backslash int\; x\backslash operatorname\{arsinh\}(ax)\backslash ,dx=\; \backslash frac\{x^2\backslash operatorname\{arsinh\}(ax)\}\{2\}+\; \backslash frac\{\backslash operatorname\{arsinh\}(ax)\}\{4a^2\}-\; \backslash frac\{x\; \backslash sqrt\{a^2x^2+1\}\}\{4a\}+C$

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

:$\backslash int\; x^m\backslash operatorname\{arsinh\}(ax)\backslash ,dx=\; \backslash frac\{x^\{m+1\}\backslash operatorname\{arsinh\}(ax)\}\{m+1\}-\; \backslash frac\{a\}\{m+1\}\backslash int\backslash frac\{x^\{m+1\}\}\{\backslash sqrt\{a^2x^2+1\}\}\backslash ,dx\backslash quad(m\backslash ne-1)$

:$\backslash int\backslash operatorname\{arsinh\}(ax)^2\backslash ,dx=\; 2x+x\backslash operatorname\{arsinh\}(ax)^2-\; \backslash frac\{2\backslash sqrt\{a^2x^2+1\}\backslash operatorname\{arsinh\}(ax)\}\{a\}+C$

:$\backslash int\backslash operatorname\{arsinh\}(ax)^n\backslash ,dx=\; x\backslash operatorname\{arsinh\}(ax)^n-\; \backslash frac\{n\backslash sqrt\{a^2x^2+1\}\backslash operatorname\{arsinh\}(ax)^\{n-1\}\}\{a\}+\; n(n-1)\backslash int\backslash operatorname\{arsinh\}(ax)^\{n-2\}\backslash ,dx$

:$\backslash int\backslash operatorname\{arsinh\}(ax)^n\backslash ,dx=\; -\backslash frac\{x\backslash operatorname\{arsinh\}(ax)^\{n+2\}\}\{(n+1)(n+2)\}+\; \backslash frac\{\backslash sqrt\{a^2x^2+1\}\backslash operatorname\{arsinh\}(ax)^\{n+1\}\}\{a(n+1)\}+\; \backslash frac\{1\}\{(n+1)(n+2)\}\backslash int\backslash operatorname\{arsinh\}(ax)^\{n+2\}\backslash ,dx\backslash quad(n\backslash ne-1,-2)$

2.2. 역쌍곡코사인 함수 (arcosh)

arcosh^영어는 역쌍곡코사인 함수이다.

:$\backslash int\backslash operatorname\{arcosh\}(a\backslash ,x)\backslash ,dx=$
x\,\operatorname{arcosh}(a\,x)-
\frac{\sqrt{a\,x+1}\,\sqrt{a\,x-1}}{a}+C

:$\backslash int\; x\backslash ,\backslash operatorname\{arcosh\}(a\backslash ,x)dx=$
\frac{x^2\,\operatorname{arcosh}(a\,x)}{2}-
\frac{\operatorname{arcosh}(a\,x)}{4\,a^2}-
\frac{x\,\sqrt{a\,x+1}\,\sqrt{a\,x-1}}{4\,a}+C

:$\backslash int\; x^2\backslash ,\backslash operatorname\{arcosh\}(a\backslash ,x)dx=$
\frac{x^3\,\operatorname{arcosh}(a\,x)}{3}-\frac{\left(a^2\,x^2+2\right)\sqrt{a\,x+1}\,\sqrt{a\,x-1}}{9\,a^3}+C

:$\backslash int\; x^m\backslash ,\backslash operatorname\{arcosh\}(a\backslash ,x)dx=$
\frac{x^{m+1}\,\operatorname{arcosh}(a\,x)}{m+1}\,-\,
\frac{a}{m+1}\int\frac{x^{m+1}}{\sqrt{a\,x+1}\,\sqrt{a\,x-1}}\,dx\quad(m\ne-1)

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

:$\backslash int\backslash operatorname\{arcosh\}(a\backslash ,x)^n\backslash ,dx=$
x\,\operatorname{arcosh}(a\,x)^n\,-\,
\frac{n\,\sqrt{a\,x+1}\,\sqrt{a\,x-1}\,\operatorname{arcosh}(a\,x)^{n-1}}{a}\,+\,
n\,(n-1)\int\operatorname{arcosh}(a\,x)^{n-2}\,dx

:$\backslash int\backslash operatorname\{arcosh\}(a\backslash ,x)^n\backslash ,dx=$
-\frac{x\,\operatorname{arcosh}(a\,x)^{n+2}}{(n+1)\,(n+2)}\,+\,
\frac{\sqrt{a\,x+1}\,\sqrt{a\,x-1}\,\operatorname{arcosh}(a\,x)^{n+1}}{a\,(n+1)}\,+\,
\frac{1}{(n+1)\,(n+2)}\int\operatorname{arcosh}(a\,x)^{n+2}\,dx\quad(n\ne-1,-2)

2.3. 역쌍곡탄젠트 함수 (artanh)

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

:$\backslash int\; x\backslash operatorname\{artanh\}(ax)\backslash ,dx=$
\frac{x^2\operatorname{artanh}(ax)}{2}-
\frac{\operatorname{artanh}(ax)}{2a^2}+\frac{x}{2a}+C

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

:$\backslash int\; x^m\backslash operatorname\{artanh\}(ax)\backslash ,dx=$
\frac{x^{m+1}\operatorname{artanh}(ax)}{m+1}-
\frac{a}{m+1}\int\frac{x^{m+1}}{1-a^2x^2}\,dx\quad(m\ne-1)

2.4. 역쌍곡코탄젠트 함수 (arcoth)

:$$\backslash int\backslash operatorname\{arcoth\}(ax)\backslash ,dx=\; x\backslash operatorname\{arcoth\}(ax)+\; \backslash frac\{\backslash ln\backslash left(a^2x^2-1\backslash right)\}\{2a\}+C$$

:$$\backslash int\; x\backslash operatorname\{arcoth\}(ax)\backslash ,dx=\; \backslash frac\{x^2\backslash operatorname\{arcoth\}(ax)\}\{2\}-\; \backslash frac\{\backslash operatorname\{arcoth\}(ax)\}\{2a^2\}+\backslash frac\{x\}\{2a\}+C$$

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

:$$\backslash int\; x^m\backslash operatorname\{arcoth\}(ax)\backslash ,dx=\; \backslash frac\{x^\{m+1\}\backslash operatorname\{arcoth\}(ax)\}\{m+1\}+\; \backslash frac\{a\}\{m+1\}\backslash int\backslash frac\{x^\{m+1\}\}\{a^2x^2-1\}\backslash ,dx\backslash quad(m\backslash ne-1)$$

2.5. 역쌍곡시컨트 함수 (arsech)

$$\backslash int\; \backslash operatorname\{arsech\}(ax)\; \backslash ,\; dx\; =\; x\; \backslash operatorname\{arsech\}(ax)\; -\; \backslash frac\{2\}\{a\}\; \backslash operatorname\{arctan\}\; \backslash sqrt\{\backslash frac\{1\; -\; ax\}\{1\; +\; ax\}\}\; +\; C$$

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

$$\backslash int\; x^2\; \backslash operatorname\{arsech\}(ax)\; \backslash ,\; dx\; =\; \backslash frac\{x^3\; \backslash operatorname\{arsech\}(ax)\}\{3\}\; -\; \backslash frac\{1\}\{3a^3\}\; \backslash operatorname\{arctan\}\; \backslash sqrt\{\backslash frac\{1\; -\; ax\}\{1\; +\; ax\}\}\; -\; \backslash frac\{x(1\; +\; ax)\}\{6a^2\}\; \backslash sqrt\{\backslash frac\{1\; -\; ax\}\{1\; +\; ax\}\}\; +\; C$$

$$\backslash int\; x^m\; \backslash operatorname\{arsech\}(ax)\; \backslash ,\; dx\; =\; \backslash frac\{x^\{m\; +\; 1\}\; \backslash operatorname\{arsech\}(ax)\}\{m\; +\; 1\}\; +\; \backslash frac\{1\}\{m\; +\; 1\}\; \backslash int\; \backslash frac\{x^m\}\{(1\; +\; ax)\; \backslash sqrt\{\backslash frac\{1\; -\; ax\}\{1\; +\; ax\}\}\}\; \backslash ,\; dx\; \backslash quad\; (m\; \backslash ne\; -1)$$

2.6. 역쌍곡코시컨트 함수 (arcsch)

:$\backslash int\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$
x\operatorname{arcsch}(ax)+
\frac{1}{a}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+C

:$\backslash int\; x\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$
\frac{x^2\operatorname{arcsch}(ax)}{2}+
\frac{x}{2a}\sqrt{\frac{1}{a^2x^2}+1}+C

:$\backslash int\; x^2\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$
\frac{x^3\operatorname{arcsch}(ax)}{3}-
\frac{1}{6a^3}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+
\frac{x^2}{6a}\sqrt{\frac{1}{a^2x^2}+1}+C

:$\backslash int\; x^m\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$
\frac{x^{m+1}\operatorname{arcsch}(ax)}{m+1}+
\frac{1}{a(m+1)}\int\frac{x^{m-1}}{\sqrt{\frac{1}{a^2x^2}+1}}\,dx\quad(m\ne-1)