## Encyclopedia > List of integrals of hyperbolic functions

Article Content

# List of integrals of hyperbolic functions

The following is a list of Integrals (Antiderivative functions) of hyperbolic functions. For a complete list of Integral functions, please see Table of Integrals and List of integrals.

$\int\sinh cx\,dx = \frac{1}{c}\cosh cx$

$\int\cosh cx\,dx = \frac{1}{c}\sinh cx$

$\int\sinh^2 cx\,dx = \frac{1}{4c}\sinh 2cx - \frac{x}{2}$

$\int\cosh^2 cx\,dx = \frac{1}{4c}\sinh 2cx + \frac{x}{2}$

$\int\sinh^n cx\,dx = \frac{1}{cn}\sinh^{n-1} cx\cosh cx - \frac{n-1}{n}\int\sinh^{n-2} cx\,dx \qquad\mbox{(for }n>0\mbox{)}$

also: $\int\sinh^n cx\,dx = \frac{1}{c(n+1)}\sinh^{n+1} cx\cosh cx - \frac{n+2}{n+1}\int\sinh^{n+2}cx\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}$

$\int\cosh^n cx\,dx = \frac{1}{cn}\sinh cx\cosh^{n-1} cx + \frac{n-1}{n}\int\cosh^{n-2} cx\,dx \qquad\mbox{(for }n>0\mbox{)}$

also: $\int\cosh^n cx\,dx = -\frac{1}{c(n+1)}\sinh cx\cosh^{n+1} cx - \frac{n+2}{n+1}\int\cosh^{n+2}cx\,dx \qquad\mbox{(for }n<0\mbox{, }n\neq -1\mbox{)}$

$\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\tanh\frac{cx}{2}\right|$

also: $\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\sinh cx}\right|$

also: $\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\sinh cx}{\cosh cx + 1}\right|$

also: $\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\cosh cx + 1}\right|$

$\int\frac{dx}{\cosh cx} = \frac{2}{c} \arctan e^{cx}$

$\int\frac{dx}{\sinh^n cx} = \frac{\cosh cx}{c(n-1)\sinh^{n-1} cx}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} cx} \qquad\mbox{(for }n\neq 1\mbox{)}$

$\int\frac{dx}{\cosh^n cx} = \frac{\sinh cx}{c(n-1)\cosh^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} cx} \qquad\mbox{(for }n\neq 1\mbox{)}$

$\int\frac{\cosh^n cx}{\sinh^m cx} dx = \frac{\cosh^{n-1} cx}{c(n-m)\sinh^{m-1} cx} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} cx}{\sinh^m cx} dx \qquad\mbox{(for }m\neq n\mbox{)}$

also: $\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n+1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-m+2}{m-1}\int\frac{\cosh^n cx}{\sinh^{m-2} cx} dx \qquad\mbox{(for }m\neq 1\mbox{)}$

also: $\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n-1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} cx}{\sinh^{m-2} cx} dx \qquad\mbox{(for }m\neq 1\mbox{)}$

$\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m-1} cx}{c(m-n)\cosh^{n-1} cx} + \frac{m-1}{m-n}\int\frac{\sinh^{m-2} cx}{\cosh^n cx} dx \qquad\mbox{(for }m\neq n\mbox{)}$

also: $\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m+1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-n+2}{n-1}\int\frac{\sinh^m cx}{\cosh^{n-2} cx} dx \qquad\mbox{(for }n\neq 1\mbox{)}$

also: $\int\frac{\sinh^m cx}{\cosh^n cx} dx = -\frac{\sinh^{m-1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} cx}{\cosh^{n-2} cx} dx \qquad\mbox{(for }n\neq 1\mbox{)}$

$\int x\sinh cx\,dx = \frac{1}{c} x\cosh cx - \frac{1}{c^2}\sinh cx$

$\int x\cosh cx\,dx = \frac{1}{c} x\sinh cx - \frac{1}{c^2}\cosh cx$

$\int \tanh cx\,dx = \frac{1}{c}\ln|\cosh cx|$

$\int \coth cx\,dx = \frac{1}{c}\ln|\sinh cx|$

$\int \tanh^n cx\,dx = -\frac{1}{c(n-1)}\tanh^{n-1} cx+\int\tanh^{n-2} cx\,dx \qquad\mbox{(for }n\neq 1\mbox{)}$

$\int \coth^n cx\,dx = -\frac{1}{c(n-1)}\coth^{n-1} cx+\int\coth^{n-2} cx\,dx \qquad\mbox{(for }n\neq 1\mbox{)}$

$\int \sinh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh cx \cosh bx - c\cosh cx \sinh bx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}$

$\int \cosh bx \cosh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \cosh cx - c\sinh cx \cosh bx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}$

$\int \cosh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \sinh cx - c\cosh bx \cosh cx) \qquad\mbox{(for }b^2\neq c^2\mbox{)}$

$\int \sinh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)$

$\int \sinh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)$

$\int \cosh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)$

$\int \cosh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)$

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Rameses ... Rameses, is the name of several Egyptian pharaohs: Ramses I[?] Ramses II ("The Great") Ramses III Ramses IV[?] The name means "Child of the Sun". This is a ...