Az alábbi lista a hiperbolikus függvények integráljait tartalmazza. Feltételezzük, hogy a c konstans nem zéró.
![{\displaystyle \int {\text{sh}}(cx)dx={\frac {1}{c}}{\text{ch}}(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e383ab36cd9668b14e70ef037c522e2252fe5e6a)
![{\displaystyle \int {\text{ch}}(cx)dx={\frac {1}{c}}{\text{sh}}(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/670fe087e636a0693544e5d75e1f5ff803b85904)
![{\displaystyle \int {\text{sh}}^{2}(cx)dx={\frac {1}{4c}}{\text{sh}}(2cx)-{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d626cf28a839594e68bda17effe102c1c88ea353)
![{\displaystyle \int {\text{ch}}^{2}(cx)dx={\frac {1}{4c}}{\text{sh}}(2cx)+{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42eb85d6cf7ff55f7b256e27d6d73e5a5d57077c)
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ed2d851ba1abf38910febd296700ccd2b5945e)
- továbbá:
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1facd0f822bf3a1e6db58696094c850123fc7c0d)
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b0a162c4af3000e621e33871e9afdefd5b552d)
- továbbá:
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0f26e172a99291ef286717f1c8ad831de41fb7)
![{\displaystyle \int {\frac {dx}{{\text{sh}}(cx)}}={\frac {1}{c}}\ln \left|{\text{th}}{\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cee2fcabd1ef15e13755e48596d2a30a7b6ebc7)
- továbbá:
![{\displaystyle \int {\frac {dx}{{\text{sh}}(cx)}}={\frac {1}{c}}\ln \left|{\frac {{\text{ch}}(cx)-1}{{\text{sh}}(cx)}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9aff468483c837ad519ec4fe2c76782dd37ecd5)
- továbbá:
![{\displaystyle \int {\frac {dx}{{\text{sh}}(cx)}}={\frac {1}{c}}\ln \left|{\frac {{\text{sh}}(cx)}{{\text{ch}}(cx)+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/037ea9a8f104d5fc48dcc682d41aa1df5ff1037f)
- továbbá:
![{\displaystyle \int {\frac {dx}{{\text{sh}}(cx)}}={\frac {1}{c}}\ln \left|{\frac {{\text{ch}}(cx)-1}{{\text{ch}}(cx)+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0c12d65e9db599ed5fa1ec3bae16c76e434d9b)
![{\displaystyle \int {\frac {dx}{{\text{ch}}(cx)}}={\frac {2}{c}}{\text{arc tg}}(e^{cx})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06a6a0dc9f2cb3f4b0a5d86e2e0a9561f54faddc)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75d4f43cb33dcb9bf51d41ef1e80902316df4bfb)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f46cad6bc47404bc1289a2389a8d841496c9c83e)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b57709991c6fb1126c84207511224f3d63d440da)
- továbbá:
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf1edfa7b2d804398b831b6808e2ad61ee523cb)
- továbbá:
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1568504e09a38fae73f772102d9afd1b1c51bd23)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab75686e395c1a5441d433ba0461187670761a5)
- továbbá:
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8023a648118849ef18f88c004dcbe2bf641445)
- továbbá:
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b32ae8d2610a325f830b979fcaaa9632018b42)
![{\displaystyle \int x\,{\text{sh}}(cx)dx={\frac {1}{c}}x\,{\text{ch}}(cx)-{\frac {1}{c^{2}}}{\text{sh}}(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebef69d7968ffc2d68d240eacd0e8fd0dbadbfb)
![{\displaystyle \int x\,{\text{ch}}(cx)dx={\frac {1}{c}}x\,{\text{sh}}(cx)-{\frac {1}{c^{2}}}{\text{ch}}(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8831e4ede532b954a60d66bd8ae7a5cb341e32bb)
![{\displaystyle \int {\text{th}}(cx)dx={\frac {1}{c}}\ln |{\text{ch}}(cx)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16e2880bb2d25a4f976ecaf09e8befdbd4217602)
![{\displaystyle \int {\text{cth}}(cx)dx={\frac {1}{c}}\ln |{\text{sh}}(cx)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3080fa17664b292cba7124e623dd0531805ce6df)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74eb54f500687ccb6c8598c8649afafe233d4c43)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2998e96d85a5ea26c8a007a0fabf5ec875eebfee)
![{\displaystyle \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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dae10deba9b874ffcab165a6598cdff94096b4)
![{\displaystyle \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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee6f820b17db979085edc8257d10b162d09d864)
![{\displaystyle \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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbbcb9bef32913f6d243bf5cbab6d12a5c0394d)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59d266bcdb005421fd6ce8e8e61cc9e790b1574)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/343eb16c6f0e095496b97e7508536ddad9e1ee03)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4dfbd8367cbe88b4aaa75263da82a2f0131343)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/599b2762772d0b45475fe03881f90a5076281102)
Matematikaportál • összefoglaló, színes tartalomajánló lap