다음은 쌍곡 함수(hyperbolic function)의 적분(integral) (역도함수(anti-derivative))의 목록입니다. 적분 함수의 전체 목록에 대해, 적분의 목록(list of integrals)을 참조하십시오.
모든 공식에서, 상수 a는 비-영으로 가정되고 C는 적분의 상수(constant of integration)를 나타냅니다.
Integrals involving only hyperbolic sine functions
\(\displaystyle \int\sinh ax\,dx = \frac{1}{a}\cosh ax+C\)
\(\displaystyle \int\sinh^2 ax\,dx = \frac{1}{4a}\sinh 2ax - \frac{x}{2}+C\)
\(\displaystyle \int\sinh^n ax\,dx = \frac{1}{an}(\sinh^{n-1} ax)(\cosh ax) - \frac{n-1}{n}\int\sinh^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\)
also: \(\displaystyle \int\sinh^n ax\,dx = \frac{1}{a(n+1)}(\sinh^{n+1} ax)(\cosh ax) - \frac{n+2}{n+1}\int\sinh^{n+2}ax\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}\)
\(\displaystyle \int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C\)
also: \(\displaystyle \int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\frac{\cosh ax - 1}{\sinh ax}\right|+C\)
\(\displaystyle \int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\frac{\sinh ax}{\cosh ax + 1}\right|+C\)
\(\displaystyle \int\frac{dx}{\sinh ax} = \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C\)
\(\displaystyle \int\frac{dx}{\sinh^n ax} = -\frac{\cosh ax}{a(n-1)\sinh^{n-1} ax}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int x\sinh ax\,dx = \frac{1}{a} x\cosh ax - \frac{1}{a^2}\sinh ax+C\)
\(\displaystyle \int (\sinh ax)(\sinh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh bx)(\cosh ax) - b(\cosh bx)(\sinh ax)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\)
Integrals involving only hyperbolic cosine functions
\(\displaystyle \int\cosh ax\,dx = \frac{1}{a}\sinh ax+C\)
\(\displaystyle \int\cosh^2 ax\,dx = \frac{1}{4a}\sinh 2ax + \frac{x}{2}+C\)
\(\displaystyle \int\cosh^n ax\,dx = \frac{1}{an}(\sinh ax)(\cosh^{n-1} ax) + \frac{n-1}{n}\int\cosh^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\)
also: \(\displaystyle \int\cosh^n ax\,dx = -\frac{1}{a(n+1)}(\sinh ax)(\cosh^{n+1} ax) + \frac{n+2}{n+1}\int\cosh^{n+2}ax\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}\)
\(\displaystyle \int\frac{dx}{\cosh ax} = \frac{2}{a} \arctan e^{ax}+C\)
also: \(\displaystyle \int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\sinh ax)+C\)
\(\displaystyle \int\frac{dx}{\cosh^n ax} = \frac{\sinh ax}{a(n-1)\cosh^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int x\cosh ax\,dx = \frac{1}{a} x\sinh ax - \frac{1}{a^2}\cosh ax+C\)
\(\displaystyle \int x^2 \cosh ax\,dx = -\frac{2x \cosh ax}{a^2} + \left(\frac{x^2}{a}+\frac{2}{a^3}\right) \sinh ax+C\)
\(\displaystyle \int (\cosh ax)(\cosh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh ax)(\cosh bx) - b(\sinh bx)(\cosh ax)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\)
\(\displaystyle \int \frac{dx}{1+\cosh(ax)} = \frac{2}{a} \frac{1}{1+e^{-ax}}+C\) or \(\displaystyle \frac{2}{a}\) times [[Logistic function|The Logistic Function]]
Other integrals
Integrals of hyperbolic tangent, cotangent, secant, cosecant functions
\(\displaystyle \int \tanh x \, dx = \ln \cosh x + C\)
\(\displaystyle \int\tanh^2 ax\,dx = x - \frac{\tanh ax}{a}+C\)
\(\displaystyle \int \tanh^n ax\,dx = -\frac{1}{a(n-1)}\tanh^{n-1} ax+\int\tanh^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 \)
\(\displaystyle \int \coth^n ax\,dx = -\frac{1}{a(n-1)}\coth^{n-1} ax+\int\coth^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C\)
\(\displaystyle \int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 \)
Integrals involving hyperbolic sine and cosine functions
\(\displaystyle \int (\cosh ax)(\sinh bx)\,dx = \frac{1}{a^2-b^2} \big(a(\sinh ax)(\sinh bx) - b(\cosh ax)(\cosh bx)\big)+C \qquad\mbox{(for }a^2\neq b^2\mbox{)}\)
\(\displaystyle \int\frac{\cosh^n ax}{\sinh^m ax} dx = \frac{\cosh^{n-1} ax}{a(n-m)\sinh^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} ax}{\sinh^m ax} dx \qquad\mbox{(for }m\neq n\mbox{)}\)
also: \(\displaystyle \int\frac{\cosh^n ax}{\sinh^m ax} dx = -\frac{\cosh^{n+1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-m+2}{m-1}\int\frac{\cosh^n ax}{\sinh^{m-2} ax} dx \qquad\mbox{(for }m\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\cosh^n ax}{\sinh^m ax} dx = -\frac{\cosh^{n-1} ax}{a(m-1)\sinh^{m-1} ax} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} ax}{\sinh^{m-2} ax} dx \qquad\mbox{(for }m\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\sinh^m ax}{\cosh^n ax} dx = \frac{\sinh^{m-1} ax}{a(m-n)\cosh^{n-1} ax} + \frac{m-1}{n-m}\int\frac{\sinh^{m-2} ax}{\cosh^n ax} dx \qquad\mbox{(for }m\neq n\mbox{)}\)
\(\displaystyle \int\frac{\sinh^m ax}{\cosh^n ax} dx = \frac{\sinh^{m+1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-n+2}{n-1}\int\frac{\sinh^m ax}{\cosh^{n-2} ax} dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\sinh^m ax}{\cosh^n ax} dx = -\frac{\sinh^{m-1} ax}{a(n-1)\cosh^{n-1} ax} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} ax}{\cosh^{n-2} ax} dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
Integrals involving hyperbolic and trigonometric functions
\(\displaystyle \int \sinh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+C\)
\(\displaystyle \int \sinh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)+C\)
\(\displaystyle \int \cosh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+C\)
\(\displaystyle \int \cosh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)+C\)