역쌍곡선함수 적분표

아래 목록은 역쌍곡선함수의 부정적분이다. 아래의 적분식들에서 상수 *a*는 0이 아니며, *C*는 적분상수이다. 각 역쌍곡선함수의 적분식은 그에 대응하는 역삼각함수 적분식이 존재한다.
# 역쌍곡사인함수의 적분
\int\operatorname{arsinh}(ax)\,dx= x\operatorname{arsinh}(ax)-\frac{\sqrt{a^2x^2+1{a}+C \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 \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 \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) \int\operatorname{arsinh}(ax)^2\,dx= 2x+x\operatorname{arsinh}(ax)^2- \frac{2\sqrt{a^2x^2+1}\operatorname{arsinh}(ax)}{a}+C \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 \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)
# 역쌍곡코사인함수의 적분
\int\operatorname{arcosh}(ax)\,dx= x\operatorname{arcosh}(ax)- \frac{\sqrt{ax+1}\sqrt{ax-1{a}+C \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 \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 \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) \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 \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 \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)
# 역쌍곡탄젠트함수의 적분
\int\operatorname{artanh}(ax)\,dx= x\operatorname{artanh}(ax)+ \frac{\ln\left(1-a^2x^2\right)}{2a}+C \int x\operatorname{artanh}(ax)\,dx= \frac{x^2\operatorname{artanh}(ax)}{2}- \frac{\operatorname{artanh}(ax)}{2a^2}+\frac{x}{2a}+C \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 \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)
# 역쌍곡코시컨트함수의 적분
\int\operatorname{arcsch}(ax)\,dx= x\operatorname{arcsch}(ax)+ \frac{1}{a}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+C \int x\operatorname{arcsch}(ax)\,dx= \frac{x^2\operatorname{arcsch}(ax)}{2}+ \frac{x}{2a}\sqrt{\frac{1}{a^2x^2}+1}+C \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 \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)
# 역쌍곡시컨트함수의 적분
\int\operatorname{arsech}(ax)\,dx= x\operatorname{arsech}(ax)- \frac{2}{a}\operatorname{arctan}\sqrt{\frac{1-ax}{1+ax+C \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 \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 \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)
# 역쌍곡코탄젠트함수의 적분
\int\operatorname{arcoth}(ax)\,dx= x\operatorname{arcoth}(ax)+ \frac{\ln\left(a^2x^2-1\right)}{2a}+C \int x\operatorname{arcoth}(ax)\,dx= \frac{x^2\operatorname{arcoth}(ax)}{2}- \frac{\operatorname{arcoth}(ax)}{2a^2}+\frac{x}{2a}+C \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 \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)