Список интегралов от гиперболических функций

Список интегралов (первообразных функций) от гиперболических функций. В списке везде опущена аддитивная константа интегрирования.

<math>\int\operatorname{sh} cx\,dx = \frac{1}{c}\operatorname{ch} cx</math>

<math>\int\operatorname{ch} cx\,dx = \frac{1}{c}\operatorname{sh} cx</math>

<math>\int\operatorname{sh}^2 cx\,dx = \frac{1}{4c}\operatorname{sh} 2cx - \frac{x}{2}</math>

<math>\int\operatorname{ch}^2 cx\,dx = \frac{1}{4c}\operatorname{sh} 2cx + \frac{x}{2}</math>

<math>\int\operatorname{sh}^n cx\,dx = \frac{1}{cn}\operatorname{sh}^{n-1} cx\operatorname{ch} cx - \frac{n-1}{n}\int\operatorname{sh}^{n-2} cx\,dx \qquad\mbox{( }n>0\mbox{)}</math>

также: <math>\int\operatorname{sh}^n cx\,dx = \frac{1}{c(n+1)}\operatorname{sh}^{n+1} cx\operatorname{ch} cx - \frac{n+2}{n+1}\int\operatorname{sh}^{n+2}cx\,dx \qquad\mbox{( }n<0\mbox{, }n\neq -1\mbox{)}</math>

<math>\int\operatorname{ch}^n cx\,dx = \frac{1}{cn}\operatorname{sh} cx\operatorname{ch}^{n-1} cx + \frac{n-1}{n}\int\operatorname{ch}^{n-2} cx\,dx \qquad\mbox{( }n>0\mbox{)}</math>

также: <math>\int\operatorname{ch}^n cx\,dx = -\frac{1}{c(n+1)}\operatorname{sh} cx\operatorname{ch}^{n+1} cx - \frac{n+2}{n+1}\int\operatorname{ch}^{n+2}cx\,dx \qquad\mbox{(}n<0\mbox{, }n\neq -1\mbox{)}</math>

<math>\int\frac{dx}{\operatorname{sh} cx} = \frac{1}{c} \ln\left|\operatorname{th}\frac{cx}{2}\right| = \frac{1}{c} \ln\left|\frac{\operatorname{ch} cx - 1}{\operatorname{sh} cx}\right| = \frac{1}{c} \ln\left|\frac{\operatorname{sh} cx}{\operatorname{ch} cx + 1}\right| = \frac{1}{c} \ln\left|\frac{\operatorname{ch} cx - 1}{\operatorname{ch} cx + 1}\right|</math>

<math>\int\frac{dx}{\operatorname{sh}^2 cx} = - \frac{1}{c}\operatorname{cth} cx</math>

<math>\int\frac{dx}{\operatorname{ch} cx} = \frac{2}{c} \operatorname{arctg} e^{cx}</math>

<math>\int\frac{dx}{\operatorname{ch}^2 cx} = \frac{1}{c}\operatorname{th} cx</math>

<math>\int\frac{dx}{\operatorname{sh}^n cx} = \frac{\operatorname{ch} cx}{c(n-1)\operatorname{sh}^{n-1} cx}-\frac{n-2}{n-1}\int\frac{dx}{\operatorname{sh}^{n-2} cx} \qquad\mbox{( }n\neq 1\mbox{)}</math>

<math>\int\frac{dx}{\operatorname{ch}^n cx} = \frac{\operatorname{sh} cx}{c(n-1)\operatorname{ch}^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\operatorname{ch}^{n-2} cx} \qquad\mbox{( }n\neq 1\mbox{)}</math>

<math>\int\frac{\operatorname{ch}^n cx}{\operatorname{sh}^m cx} dx = \frac{\operatorname{ch}^{n-1} cx}{c(n-m)\operatorname{sh}^{m-1} cx} + \frac{n-1}{n-m}\int\frac{\operatorname{ch}^{n-2} cx}{\operatorname{sh}^m cx} dx \qquad\mbox{( }m\neq n\mbox{)}</math>

также: <math>\int\frac{\operatorname{ch}^n cx}{\operatorname{sh}^m cx} dx = -\frac{\operatorname{ch}^{n+1} cx}{c(m-1)\operatorname{sh}^{m-1} cx} + \frac{n-m+2}{m-1}\int\frac{\operatorname{ch}^n cx}{\operatorname{sh}^{m-2} cx} dx \qquad\mbox{( }m\neq 1\mbox{)}</math>

также: <math>\int\frac{\operatorname{ch}^n cx}{\operatorname{sh}^m cx} dx = -\frac{\operatorname{ch}^{n-1} cx}{c(m-1)\operatorname{sh}^{m-1} cx} + \frac{n-1}{m-1}\int\frac{\operatorname{ch}^{n-2} cx}{\operatorname{sh}^{m-2} cx} dx \qquad\mbox{( }m\neq 1\mbox{)}</math>

<math>\int\frac{\operatorname{sh}^m cx}{\operatorname{ch}^n cx} dx = \frac{\operatorname{sh}^{m-1} cx}{c(m-n)\operatorname{ch}^{n-1} cx} + \frac{m-1}{m-n}\int\frac{\operatorname{sh}^{m-2} cx}{\operatorname{ch}^n cx} dx \qquad\mbox{( }m\neq n\mbox{)}</math>

также: <math>\int\frac{\operatorname{sh}^m cx}{\operatorname{ch}^n cx} dx = \frac{\operatorname{sh}^{m+1} cx}{c(n-1)\operatorname{ch}^{n-1} cx} + \frac{m-n+2}{n-1}\int\frac{\operatorname{sh}^m cx}{\operatorname{ch}^{n-2} cx} dx \qquad\mbox{( }n\neq 1\mbox{)}</math>

также: <math>\int\frac{\operatorname{sh}^m cx}{\operatorname{ch}^n cx} dx = -\frac{\operatorname{sh}^{m-1} cx}{c(n-1)\operatorname{ch}^{n-1} cx} + \frac{m-1}{n-1}\int\frac{\operatorname{sh}^{m -2} cx}{\operatorname{ch}^{n-2} cx} dx \qquad\mbox{( }n\neq 1\mbox{)}</math>

<math>\int x\operatorname{sh} cx\,dx = \frac{1}{c} x\operatorname{ch} cx - \frac{1}{c^2}\operatorname{sh} cx</math>

<math>\int x\operatorname{ch} cx\,dx = \frac{1}{c} x\operatorname{sh} cx - \frac{1}{c^2}\operatorname{ch} cx</math>

<math>\int \operatorname{th} cx\,dx = \frac{1}{c}\ln|\operatorname{ch} cx|</math>

<math>\int \operatorname{cth} cx\,dx = \frac{1}{c}\ln|\operatorname{sh} cx|</math>

<math>\int \operatorname{th}^2 cx\,dx = x - \frac{1}{c}\operatorname{th} cx</math>

<math>\int \operatorname{cth}^2 cx\,dx = x - \frac{1}{c}\operatorname{cth} cx</math>

<math>\int \operatorname{th}^n cx\,dx = -\frac{1}{c(n-1)}\operatorname{th}^{n-1} cx+\int\operatorname{th}^{n-2} cx\,dx \qquad\mbox{( }n\neq 1\mbox{ )}</math>

<math>\int \operatorname{cth}^n cx\,dx = -\frac{1}{c(n-1)}\operatorname{cth}^{n-1} cx+\int\operatorname{cth}^{n-2} cx\,dx \qquad\mbox{( }n\neq 1\mbox{)}</math>

<math>\int \operatorname{sh} bx \operatorname{sh} cx\,dx = \frac{1}{b^2-c^2} (b\operatorname{sh} cx \operatorname{ch} bx - c\operatorname{ch} cx \operatorname{sh} bx) \qquad\mbox{( }b^2\neq c^2\mbox{)}</math>

<math>\int \operatorname{ch} bx \operatorname{ch} cx\,dx = \frac{1}{b^2-c^2} (b\operatorname{sh} bx \operatorname{ch} cx - c\operatorname{sh} cx \operatorname{ch} bx) \qquad\mbox{( }b^2\neq c^2\mbox{)}</math>

<math>\int \operatorname{ch} bx \operatorname{sh} cx\,dx = \frac{1}{b^2-c^2} (b\operatorname{sh} bx \operatorname{sh} cx - c\operatorname{ch} bx \operatorname{ch} cx) \qquad\mbox{( }b^2\neq c^2\mbox{)}</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>