Ondorengoa funtzio hiperbolikoen integralen zerrenda bat da (jatorrizkoak edo antideribatuak). Integralen zerrenda osatuago nahi baduzu, ikusi integralen zerrenda.
K erabiltzen da integrazio-konstante gisa. Konstante hori zehaztu daiteke soilik integralaren balioa ezaguna baldin bada puntu batean. Horrela, funtzio bakoitzak jatorrizkoen kopuru infinitua dauka.
![{\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b488edb1b28fd901dfba1d17ec525a3c56c3b3e)
![{\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50e9ec89d9311ebc8db52ce7a6fc659732cb5ce1)
![{\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10a0c366314ae8fee646e9d5ceed949e78cd153d)
![{\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54803511ac67b90f9fbdb9e1fea7ca8b4f525cc0)
![{\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/413e358913026e4dfc52c1cab4a8fa0a34f7125f)
![{\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{an}}\sinh ^{n-1}ax\cosh ax-{\frac {n-1}{n}}\int \sinh ^{n-2}ax\,dx\qquad {\mbox{( }}n>0{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1589a152379195669f7083d7648c38169a80fc83)
- baita hau ere:
![{\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8053828c4aaacd34562ac7f39c5c437ae614b3e3)
![{\displaystyle \int \cosh ^{n}ax\,dx={\frac {1}{an}}\sinh ax\cosh ^{n-1}ax+{\frac {n-1}{n}}\int \cosh ^{n-2}ax\,dx\qquad {\mbox{( }}n>0{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/023373e2ede25168a7bc41188a9ff08ca95bfe3b)
- baita hau ere:
![{\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d909aee0d2bd9bc79c70db2be993afde5920e75)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f42d73f82482a36d69ca943ac98c5c12e39d3cdf)
- baita hau ere:
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbc8f74e1142a117e704bbd9d08afc1903f191a)
- baita hau ere:
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7954b6751df5cef31aaf4ee831e86c1d592088a1)
- baita hau ere:
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/782c26963fed4bd6e6cfab4ea053ee4c82f4fdf7)
![{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/346a7b4564afe65335bd6ae51330ae0ce4c8e577)
![{\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ab369ca11168c57d76dcd33062e9636f3c52d5)
![{\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/977a6b307f1fd9ebcae253bd9ea2e513efa849b0)
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3675097a96002c7643d19a79f27440e9b4df82e)
- baita hau ere:
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14d39e75db06ecc8baed9828a314eb074244f1ee)
- baita hau ere:
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c4ab871b5a90ff577a488e4c2eec1f3d591425)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf887508f54d0acb1ad9ccae141c2cc13c31a66a)
- baita hau ere:
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5720ffd1252b49f841f0c5d9e3b921fe910430ec)
- baita hau ere:
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb9a9749c14583f9cb4cdb09da6c46c09b354f4)
![{\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3d27c9c29661cf4ca535115338b52262375153)
![{\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9532f048e18b5c67649370247417daa904c882d4)
![{\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea068b480be197608bb2341dfdb3571c090c13d)
![{\displaystyle \int \tanh ax\,dx={\frac {1}{a}}\ln |\cosh ax|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bd777f77c3fad995ee59be4345f2f833dbca43)
![{\displaystyle \int \coth ax\,dx={\frac {1}{a}}\ln |\sinh ax|+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7efe87aa3f4d8f6ea6ff13766b84a0e4bda312b3)
![{\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1aa4c55269f2b1de3c558c2ccddc2bbbbcaf11)
![{\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/830abf9a597d37743b4fb867d617162a1487b9d1)
![{\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+K\qquad {\mbox{( }}a^{2}\neq b^{2}{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14221ce517834d830482801e55e7d04c59ec924e)
![{\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+K\qquad {\mbox{( }}a^{2}\neq b^{2}{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e94ddaa391da5ee3a696c69aa3b97eb842c30756)
![{\displaystyle \int \cosh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\sinh bx-b\cosh ax\cosh bx)+K\qquad {\mbox{( }}a^{2}\neq b^{2}{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4da00585a71123ad2fe64049b78d7d76b1df92)
![{\displaystyle \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)+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13d438f01de3ed256ee8ce96694a2072f1110b8c)
![{\displaystyle \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)+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/affa774d27f86f09bc6fb216ebdd3446d9d99a22)
![{\displaystyle \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)+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7076b8c95540e53ac86279767443d9804db735c)
![{\displaystyle \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)+K\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec98aaadd5e4fdf5d532471d4736e572b782389b)