# Seznam integralov hiperboličnih funkcij

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Seznam integralov hiperboličnih funkcij vsebuje integrale hiperboličnih funkcij.

V vseh obrazcih je konstanta a neničelna vrednost, C označuje aditivno konstanto.

$\int\operatorname {sh} ax\,dx = \frac{1}{a}\operatorname {ch} ax+C\,$
$\int\operatorname {ch} ax\,dx = \frac{1}{a}\operatorname {sh} ax+C\,$
$\int\operatorname {sh}^2 ax\,dx = \frac{1}{4a}\operatorname {sh} 2ax - \frac{x}{2}+C\,$
$\int\operatorname {ch}^2 ax\,dx = \frac{1}{4a}\operatorname {sh} 2ax + \frac{x}{2}+C\,$
$\int\operatorname {th}^2 ax\,dx = x - \frac{\operatorname {th} ax}{a}+C\,$
$\int\operatorname {sh}^n ax\,dx = \frac{1}{an}\operatorname {sh}^{n-1} ax\operatorname {ch} ax - \frac{n-1}{n}\int\operatorname {sh}^{n-2} ax\,dx \qquad\mbox{(za }n>0\mbox{)}\,$
tudi: $\int\operatorname {sh}^n ax\,dx = \frac{1}{a(n+1)}\operatorname {sh}^{n+1} ax\operatorname {ch} ax - \frac{n+2}{n+1}\int\operatorname{sh}^{n+2}ax\,dx \qquad\mbox{(za }n<0\mbox{, }n\neq -1\mbox{)}\,$
$\int\operatorname {ch}^n ax\,dx = \frac{1}{an}\operatorname {sh} ax\operatorname {ch}^{n-1} ax + \frac{n-1}{n}\int\operatorname {ch}^{n-2} ax\,dx \qquad\mbox{(za }n>0\mbox{)}\,$
tudi: $\int\operatorname {ch}^n ax\,dx = -\frac{1}{a(n+1)}\operatorname{sh} ax\operatorname {ch}^{n+1} ax - \frac{n+2}{n+1}\int\operatorname{ch}^{n+2}ax\,dx \qquad\mbox{(za }n<0\mbox{, }n\neq -1\mbox{)}\,$
$\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\operatorname {th}\frac{ax}{2}\right|+C\,$
tudi: $\int\frac{dx}{\operatorname {sh} ax} = \frac{1}{a} \ln\left|\frac{\operatorname {ch} ax - 1}{\operatorname {sh} ax}\right|+C\,$
tudi: $\int\frac{dx}{\operatorname {sh} ax} = \frac{1}{a} \ln\left|\frac{\operatorname {sh} ax}{\operatorname {ch} ax + 1}\right|+C\,$
tudi: $\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\frac{\operatorname {ch} ax - 1}{\operatorname {ch} ax + 1}\right|+C\,$
$\int\frac{dx}{\operatorname {ch} ax} = \frac{2}{a} \arctan e^{ax}+C\,$
tudi: $\int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\operatorname {sh} ax)+C\,$
$\int\frac{dx}{\sinh^n ax} = -\frac{\operatorname {ch} ax}{a(n-1)\operatorname {sh}^{n-1} ax}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} ax} \qquad\mbox{(za }n\neq 1\mbox{)}\,$
$\int\frac{dx}{\operatorname {ch}^n ax} = \frac{\operatorname {sh} ax}{a(n-1)\operatorname {ch}^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\operatorname {ch}^{n-2} ax} \qquad\mbox{(za }n\neq 1\mbox{)}\,$
$\int\frac{\operatorname {ch}^n ax}{\sinh^m ax} dx = \frac{\operatorname {ch}^{n-1} ax}{a(n-m)\operatorname {sh}^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\operatorname {ch}^{n-2} ax}{\operatorname {sh}^m ax} dx \qquad\mbox{(za }m\neq n\mbox{)}\,$
tudi: $\int\frac{\operatorname {ch}^n ax}{\operatorname {sh}^m ax} dx = -\frac{\operatorname {ch}^{n+1} ax}{a(m-1)\operatorname {sh}^{m-1} ax} + \frac{n-m+2}{m-1}\int\frac{\operatorname {ch}^n ax}{\operatorname {sh}^{m-2} ax} dx \qquad\mbox{(za }m\neq 1\mbox{)}\,$
tudi: $\int\frac{\operatorname {ch}^n ax}{\sinh^m ax} dx = -\frac{\operatorname {ch}^{n-1} ax}{a(m-1)\operatorname {sh}^{m-1} ax} + \frac{n-1}{m-1}\int\frac{\operatorname {ch}^{n-2} ax}{\operatorname {sh}^{m-2} ax} dx \qquad\mbox{(za }m\neq 1\mbox{)}\,$
$\int\frac{\operatorname {sh}^m ax}{\operatorname {ch}^n ax} dx = \frac{\operatorname {sh}^{m-1} ax}{a(m-n)\operatorname {ch}^{n-1} ax} + \frac{m-1}{n-m}\int\frac{\operatorname {sh}^{m-2} ax}{\operatorname {ch}^n ax} dx \qquad\mbox{(za }m\neq n\mbox{)}\,$
tudi: $\int\frac{\operatorname {sh}^m ax}{\operatorname {ch}^n ax} dx = \frac{\operatorname {sh}^{m+1} ax}{a(n-1)\operatorname {ch}^{n-1} ax} + \frac{m-n+2}{n-1}\int\frac{\operatorname {sh}^m ax}{\cosh^{n-2} ax} dx \qquad\mbox{(za }n\neq 1\mbox{)}\,$
tudi: $\int\frac{\operatorname {sh}^m ax}{\cosh^n ax} dx = -\frac{\operatorname {sh}^{m-1} ax}{a(n-1)\operatorname {ch}^{n-1} ax} + \frac{m-1}{n-1}\int\frac{\operatorname {sh}^{m -2} ax}{\operatorname {ch}^{n-2} ax} dx \qquad\mbox{(za }n\neq 1\mbox{)}\,$
$\int x\operatorname {sh} ax\,dx = \frac{1}{a} x\operatorname {ch} ax - \frac{1}{a^2}\operatorname {sh} ax+C\,$
$\int x\operatorname {ch} ax\,dx = \frac{1}{a} x\operatorname {sh} ax - \frac{1}{a^2}\operatorname {ch} ax+C\,$
$\int x^2 \operatorname {ch} ax\,dx = -\frac{2x \operatorname {ch} ax}{a^2} + \left(\frac{x^2}{a}+\frac{2}{a^3}\right) \operatorname {sh} ax+C\,$
$\int \operatorname {th} ax\,dx = \frac{1}{a}\ln\operatorname {ch} ax+C\,$
$\int \operatorname {ch} ax\,dx = \frac{1}{a}\ln|\operatorname {sh} ax|+C\,$
$\int \operatorname {th}^n ax\,dx = -\frac{1}{a(n-1)}\operatorname {th}^{n-1} ax+\int\operatorname {th}^{n-2} ax\,dx \qquad\mbox{(za }n\neq 1\mbox{)}\,$
$\int \operatorname {ch}^n ax\,dx = -\frac{1}{a(n-1)}\operatorname {ch}^{n-1} ax+\int\operatorname {ch}^{n-2} ax\,dx \qquad\mbox{(za }n\neq 1\mbox{)}\,$
$\int \operatorname {sh} ax \operatorname {sh} bx\,dx = \frac{1}{a^2-b^2} (a\operatorname {sh} bx \operatorname {ch} ax - b\operatorname {ch} bx \operatorname {sh} ax)+C \qquad\mbox{(za }a^2\neq b^2\mbox{)}\,$
$\int \operatorname {ch} ax \operatorname {ch} bx\,dx = \frac{1}{a^2-b^2} (a\operatorname {sh} ax \operatorname {ch} bx - b\operatorname {sh} bx \operatorname {ch} ax)+C \qquad\mbox{(za }a^2\neq b^2\mbox{)}\,$
$\int \operatorname {ch} ax \operatorname {sh} bx\,dx = \frac{1}{a^2-b^2} (a\operatorname {sh} ax \operatorname {sh} bx - b\operatorname {ch} ax \operatorname {ch} bx)+C \qquad\mbox{(za }a^2\neq b^2\mbox{)}\,$
$\int \operatorname {sh} (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\operatorname {ch}(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\operatorname {sh}(ax+b)\cos(cx+d)+C\,$
$\int \operatorname {sh} (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\operatorname {ch}(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\operatorname {sh}(ax+b)\sin(cx+d)+C\,$
$\int \operatorname {ch} (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\operatorname {sh}(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\operatorname {ch}(ax+b)\cos(cx+d)+C\,$
$\int \operatorname {ch} (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\operatorname {sh}(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\operatorname {ch}(ax+b)\sin(cx+d)+C\,$