Hiperbolikus függvények integráljainak listája

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Az alábbi lista a hiperbolikus függvények integráljait tartalmazza. Feltételezzük, hogy a c konstans nem zéró.

\int\text {sh} (cx)dx = \frac{1}{c}\text {ch} (cx)
\int\text {ch} (cx)dx = \frac{1}{c}\text {sh} (cx)
\int\text {sh}^2 (cx)dx = \frac{1}{4c}\text {sh} (2cx) - \frac{x}{2}
\int\text {ch}^2 (cx)dx = \frac{1}{4c}\text {sh} (2cx) + \frac{x}{2}
\int\text {sh}^n (cx)dx = \frac{1}{cn}\text {sh}^{n-1} (cx)\text {ch} (cx) - \frac{n-1}{n}\int\text {sh}^{n-2} (cx)dx \qquad (n=1,2,\dots)
továbbá: \int\text {sh}^n (cx)dx = \frac{1}{c(n+1)}\text {sh}^{n+1} (cx)\text {ch} (cx) - \frac{n+2}{n+1}\int\text {sh}^{n+2}(cx)dx \qquad (n= -2,-3,\dots)
\int\text {ch}^n (cx)dx = \frac{1}{cn}\text {sh} (cx)\text {ch}^{n-1} (cx) + \frac{n-1}{n}\int\text {ch}^{n-2} (cx)dx \qquad (n=1,2,\dots)
továbbá: \int\text {ch}^n (cx)dx = -\frac{1}{c(n+1)}\text {sh} (cx)\text {ch}^{n+1} (cx) - \frac{n+2}{n+1}\int\text {ch}^{n+2}(cx)dx \qquad (n=-2,-3,\dots)
\int\frac{dx}{\text{sh} (cx)} = \frac{1}{c} \ln\left|\text {th}\frac{cx}{2}\right|
továbbá: \int\frac{dx}{\text {sh} (cx)} = \frac{1}{c} \ln\left|\frac{\text {ch} (cx) - 1}{\text {sh} (cx)}\right|
továbbá: \int\frac{dx}{\text {sh} (cx)} = \frac{1}{c} \ln\left|\frac{\text {sh} (cx)}{\text {ch} (cx) + 1}\right|
továbbá: \int\frac{dx}{\text {sh} (cx)} = \frac{1}{c} \ln\left|\frac{\text {ch} (cx) - 1}{\text {ch} (cx) + 1}\right|
\int\frac{dx}{\text {ch} (cx)} = \frac{2}{c} \text{arc tg} (e^{cx})
\int\frac{dx}{\text {sh}^n (cx)} = \frac{\text {ch} (cx)}{c(n-1)\text {sh}^{n-1} (cx)}-\frac{n-2}{n-1}\int\frac{dx}{\text {sh}^{n-2} (cx)} \qquad (n\neq 1)
\int\frac{dx}{\text {ch}^n (cx)} = \frac{\text {sh} (cx)}{c(n-1)\text {ch}^{n-1} (cx)}+\frac{n-2}{n-1}\int\frac{dx}{\text {ch}^{n-2} (cx)} \qquad (n\neq 1)
\int\frac{\text {ch}^n (cx)}{\text {sh}^m (cx)} dx = \frac{\text {ch}^{n-1} (cx)}{c(n-m)\text {sh}^{m-1} (cx)} + \frac{n-1}{n-m}\int\frac{\text {ch}^{n-2} (cx)}{\text {sh}^m (cx)} dx \qquad (m\neq n)
továbbá: \int\frac{\text {ch}^n (cx)}{\text {sh}^m (cx)} dx = -\frac{\text {ch}^{n+1} (cx)}{c(m-1)\text {sh}^{m-1} (cx)} + \frac{n-m+2}{m-1}\int\frac{\text {ch}^n (cx)}{\text {sh}^{m-2} (cx)} dx \qquad(m\neq 1)
továbbá: \int\frac{\text {ch}^n (cx)}{\text {sh}^m (cx)} dx = -\frac{\text {ch}^{n-1} (cx)}{c(m-1)\text {sh}^{m-1} (cx)} + \frac{n-1}{m-1}\int\frac{\text {ch}^{n-2} (cx)}{\text {sh}^{m-2} (cx)} dx \qquad(m\neq 1)
\int\frac{\text {sh}^m (cx)}{\text {ch}^n (cx)} dx = \frac{\text {sh}^{m-1} (cx)}{c(m-n)\text {ch}^{n-1} (cx)} + \frac{m-1}{m-n}\int\frac{\text {sh}^{m-2} (cx)}{\text {ch}^n (cx)} dx \qquad(m\neq n)
továbbá: \int\frac{\text {sh}^m (cx)}{\text {ch}^n (cx)} dx = \frac{\text {sh}^{m+1} (cx)}{c(n-1)\text {ch}^{n-1} (cx)} + \frac{m-n+2}{n-1}\int\frac{\text {sh}^m (cx)}{\text {ch}^{n-2} (cx)} dx \qquad(n\neq 1)
továbbá: \int\frac{\text {sh}^m (cx)}{\text {ch}^n (cx)} dx = -\frac{\text {sh}^{m-1} (cx)}{c(n-1)\text {ch}^{n-1} (cx)} + \frac{m-1}{n-1}\int\frac{\text {sh}^{m -2} (cx)}{\text {ch}^{n-2} (cx)} dx \qquad(n\neq 1)
\int x\,\text {sh} (cx)dx = \frac{1}{c} x\,\text {ch} (cx) - \frac{1}{c^2}\text {sh} (cx)
\int x\,\text {ch} (cx)dx = \frac{1}{c} x\,\text {sh} (cx) - \frac{1}{c^2}\text {ch} (cx)
\int \text {th} (cx)dx = \frac{1}{c}\ln|\text {ch} (cx)|
\int \text {cth} (cx)dx = \frac{1}{c}\ln|\text {sh} (cx)|
\int \text {th}^n (cx)dx = -\frac{1}{c(n-1)}\text {th}^{n-1} (cx)+\int\text {th}^{n-2} (cx)dx \qquad(n\neq 1)
\int \text {cth}^n (cx)dx = -\frac{1}{c(n-1)}\text {cth}^{n-1} (cx)+\int\text {cth}^{n-2} (cx)dx \qquad(n\neq 1)
\int \text {sh} (bx) \text {sh} (cx)dx = \frac{1}{b^2-c^2} \left(b\,\text {sh} (cx) \text {ch} (bx) - c\,\text {ch} (cx) \text {sh} (bx)\right) \qquad (b^2\neq c^2)
\int \text {ch} (bx) \text {ch} (cx)dx = \frac{1}{b^2-c^2} (b\,\text {sh} (bx) \text {ch} (cx) - c\,\text {sh} (cx) \text {ch} (bx)) \qquad (b^2\neq c^2)
\int \text {ch} (bx) \text {sh} (cx)dx = \frac{1}{b^2-c^2} (b\,\text {sh} (bx) \text {sh} (cx) - c\,\text {ch} (bx) \text {ch} (cx)) \qquad (b^2\neq c^2)
\int \text {sh} (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\text {ch}(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\text {sh}(ax+b)\cos(cx+d)
\int \text {sh} (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\text {ch}(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\text {sh}(ax+b)\sin(cx+d)
\int \text {ch} (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\text {sh}(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\text {ch}(ax+b)\cos(cx+d)
\int \text {ch} (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\text {sh}(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\text {ch}(ax+b)\sin(cx+d)