{{위키데이터 속성 추적}}
아래 목록은 [[쌍곡선 함수|역쌍곡선함수]]의 [[부정적분]]이다.

아래의 적분식들에서 상수 ''a''는 0이 아니며, ''C''는 [[적분상수]]이다. 각 역쌍곡선함수의 적분식은 그에 대응하는 [[역삼각함수 적분표|역삼각함수 적분식]]이 존재한다.

== 역쌍곡사인함수의 적분 ==
<math display="block">\int\operatorname{arsinh}(ax)\,dx=
  x\operatorname{arsinh}(ax)-\frac{\sqrt{a^2x^2+1}}{a}+C</math>

<math display="block">\int x\operatorname{arsinh}(ax)\,dx=
  \frac{x^2\operatorname{arsinh}(ax)}{2}+
  \frac{\operatorname{arsinh}(ax)}{4a^2}-
  \frac{x \sqrt{a^2x^2+1}}{4a}+C</math>

<math display="block">\int x^2\operatorname{arsinh}(ax)\,dx=
  \frac{x^3\operatorname{arsinh}(ax)}{3}-
  \frac{\left(a^2x^2-2\right)\sqrt{a^2x^2+1}}{9a^3}+C</math>

<math display="block">\int x^m\operatorname{arsinh}(ax)\,dx=
  \frac{x^{m+1}\operatorname{arsinh}(ax)}{m+1}-
  \frac{a}{m+1}\int\frac{x^{m+1}}{\sqrt{a^2x^2+1}}\,dx\quad(m\ne-1)</math>

<math display="block">\int\operatorname{arsinh}(ax)^2\,dx=
  2x+x\operatorname{arsinh}(ax)^2-
  \frac{2\sqrt{a^2x^2+1}\operatorname{arsinh}(ax)}{a}+C</math>

<math display="block">\int\operatorname{arsinh}(ax)^n\,dx=
  x\operatorname{arsinh}(ax)^n-
  \frac{n\sqrt{a^2x^2+1}\operatorname{arsinh}(ax)^{n-1}}{a}+
  n(n-1)\int\operatorname{arsinh}(ax)^{n-2}\,dx</math>

<math display="block">\int\operatorname{arsinh}(ax)^n\,dx=
  -\frac{x\operatorname{arsinh}(ax)^{n+2}}{(n+1)(n+2)}+
  \frac{\sqrt{a^2x^2+1}\operatorname{arsinh}(ax)^{n+1}}{a(n+1)}+
  \frac{1}{(n+1)(n+2)}\int\operatorname{arsinh}(ax)^{n+2}\,dx\quad(n\ne-1,-2)</math>

== 역쌍곡코사인함수의 적분 ==
<math display="block">\int\operatorname{arcosh}(ax)\,dx=
  x\operatorname{arcosh}(ax)-
  \frac{\sqrt{ax+1}\sqrt{ax-1}}{a}+C</math>

<math display="block">\int x\operatorname{arcosh}(ax)\,dx=
  \frac{x^2\operatorname{arcosh}(ax)}{2}-
  \frac{\operatorname{arcosh}(ax)}{4a^2}-
  \frac{x\sqrt{ax+1}\sqrt{ax-1}}{4a}+C</math>

<math display="block">\int x^2\operatorname{arcosh}(ax)\,dx=
  \frac{x^3\operatorname{arcosh}(ax)}{3}-\frac{\left(a^2x^2+2\right)\sqrt{ax+1}\sqrt{ax-1}}{9a^3}+C</math>

<math display="block">\int x^m\operatorname{arcosh}(ax)\,dx=
  \frac{x^{m+1}\operatorname{arcosh}(ax)}{m+1}-
  \frac{a}{m+1}\int\frac{x^{m+1}}{\sqrt{ax+1}\sqrt{ax-1}}\,dx\quad(m\ne-1)</math>

<math display="block">\int\operatorname{arcosh}(ax)^2\,dx=
  2x+x\operatorname{arcosh}(ax)^2-
  \frac{2\sqrt{ax+1}\sqrt{ax-1}\operatorname{arcosh}(ax)}{a}+C</math>

<math display="block">\int\operatorname{arcosh}(ax)^n\,dx=
  x\operatorname{arcosh}(ax)^n-
  \frac{n\sqrt{ax+1}\sqrt{ax-1}\operatorname{arcosh}(ax)^{n-1}}{a}+
  n(n-1)\int\operatorname{arcosh}(ax)^{n-2}\,dx</math>

<math display="block">\int\operatorname{arcosh}(ax)^n\,dx=
  -\frac{x\operatorname{arcosh}(ax)^{n+2}}{(n+1)(n+2)}+
  \frac{\sqrt{ax+1}\sqrt{ax-1}\operatorname{arcosh}(ax)^{n+1}}{a(n+1)}+
  \frac{1}{(n+1)(n+2)}\int\operatorname{arcosh}(ax)^{n+2}\,dx\quad(n\ne-1,-2)</math>

== 역쌍곡탄젠트함수의 적분 ==
<math display="block">\int\operatorname{artanh}(ax)\,dx=
  x\operatorname{artanh}(ax)+
  \frac{\ln\left(1-a^2x^2\right)}{2a}+C</math>

<math display="block">\int x\operatorname{artanh}(ax)\,dx=
  \frac{x^2\operatorname{artanh}(ax)}{2}-
  \frac{\operatorname{artanh}(ax)}{2a^2}+\frac{x}{2a}+C</math>

<math display="block">\int x^2\operatorname{artanh}(ax)\,dx=
  \frac{x^3\operatorname{artanh}(ax)}{3}+
  \frac{\ln\left(1-a^2x^2\right)}{6a^3}+\frac{x^2}{6a}+C</math>

<math display="block">\int x^m\operatorname{artanh}(ax)\,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)</math>

== 역쌍곡코시컨트함수의 적분 ==
<math display="block">\int\operatorname{arcsch}(ax)\,dx=
  x\operatorname{arcsch}(ax)+
  \frac{1}{a}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+C</math>

<math display="block">\int x\operatorname{arcsch}(ax)\,dx=
  \frac{x^2\operatorname{arcsch}(ax)}{2}+
  \frac{x}{2a}\sqrt{\frac{1}{a^2x^2}+1}+C</math>

<math display="block">\int x^2\operatorname{arcsch}(ax)\,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</math>

<math display="block">\int x^m\operatorname{arcsch}(ax)\,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)</math>

== 역쌍곡시컨트함수의 적분 ==
<math display="block">\int\operatorname{arsech}(ax)\,dx=
  x\operatorname{arsech}(ax)-
  \frac{2}{a}\operatorname{arctan}\sqrt{\frac{1-ax}{1+ax}}+C</math>

<math display="block">\int x\operatorname{arsech}(ax)\,dx=
  \frac{x^2\operatorname{arsech}(ax)}{2}-
  \frac{(1+ax)}{2a^2}\sqrt{\frac{1-ax}{1+ax}}+C</math>

<math display="block">\int x^2\operatorname{arsech}(ax)\,dx=
  \frac{x^3\operatorname{arsech}(ax)}{3}-
  \frac{1}{3a^3}\operatorname{arctan}\sqrt{\frac{1-ax}{1+ax}}-
  \frac{x(1+ax)}{6a^2}\sqrt{\frac{1-ax}{1+ax}}+C</math>

<math display="block">\int x^m\operatorname{arsech}(ax)\,dx=
  \frac{x^{m+1}\operatorname{arsech}(ax)}{m+1}+
  \frac{1}{m+1}\int\frac{x^m}{(1+ax)\sqrt{\frac{1-ax}{1+ax}}}\,dx\quad(m\ne-1)</math>

== 역쌍곡코탄젠트함수의 적분 ==
<math display="block">\int\operatorname{arcoth}(ax)\,dx=
  x\operatorname{arcoth}(ax)+
  \frac{\ln\left(a^2x^2-1\right)}{2a}+C</math>

<math display="block">\int x\operatorname{arcoth}(ax)\,dx=
  \frac{x^2\operatorname{arcoth}(ax)}{2}-
  \frac{\operatorname{arcoth}(ax)}{2a^2}+\frac{x}{2a}+C</math>

<math display="block">\int x^2\operatorname{arcoth}(ax)\,dx=
  \frac{x^3\operatorname{arcoth}(ax)}{3}+
  \frac{\ln\left(a^2x^2-1\right)}{6a^3}+\frac{x^2}{6a}+C</math>

<math display="block">\int x^m\operatorname{arcoth}(ax)\,dx=
  \frac{x^{m+1}\operatorname{arcoth}(ax)}{m+1}+
  \frac{a}{m+1}\int\frac{x^{m+1}}{a^2x^2-1}\,dx\quad(m\ne-1)</math>

== 같이 보기 ==
* [[적분표]]
* [[쌍곡선 함수]]

[[분류:적분]]