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Seznam integralov hiperboličnih funkcij vsebuje integrale hiperboličnih funkcij.
V vseh obrazcih je konstanta a neničelna vrednost, C
označuje aditivno konstanto.
![{\displaystyle \int \operatorname {sh} ax\,dx={\frac {1}{a}}\operatorname {ch} ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de805540d26bafd69c617862fd93231240959178)
![{\displaystyle \int \operatorname {ch} ax\,dx={\frac {1}{a}}\operatorname {sh} ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4cd1b777bc2ffe81472297aca45f5f187aa346c)
![{\displaystyle \int \operatorname {sh} ^{2}ax\,dx={\frac {1}{4a}}\operatorname {sh} 2ax-{\frac {x}{2}}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c73494884427dbbcff2431ac4e278cb2202019)
![{\displaystyle \int \operatorname {ch} ^{2}ax\,dx={\frac {1}{4a}}\operatorname {sh} 2ax+{\frac {x}{2}}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d894145c7429370fd8af03d0b496d92d242a9e60)
![{\displaystyle \int \operatorname {th} ^{2}ax\,dx=x-{\frac {\operatorname {th} ax}{a}}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16e3007a10d34abbf735bb1334a370832d0bbf2c)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20c17757e463b0cbb47f81c67b63bcd1a338a4a5)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e96e2dcb20b2d33a0f147a95c96a696848f44f5)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31857e899f250834560b994b95c99247000beb73)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/151c54705978bb6de3b787cbe1d9e59649b5ebf4)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\operatorname {th} {\frac {ax}{2}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c55e1fc09a2d2c908ee5e581ede356c4a05fffe)
- tudi:
![{\displaystyle \int {\frac {dx}{\operatorname {sh} ax}}={\frac {1}{a}}\ln \left|{\frac {\operatorname {ch} ax-1}{\operatorname {sh} ax}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f10d4f49c694f29c7dd8a82af39e8bab8dca3e6)
- tudi:
![{\displaystyle \int {\frac {dx}{\operatorname {sh} ax}}={\frac {1}{a}}\ln \left|{\frac {\operatorname {sh} ax}{\operatorname {ch} ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/939124a434cbf73bb08b599837e7d9b0d444cab4)
- tudi:
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\operatorname {ch} ax-1}{\operatorname {ch} ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b55352b2e4d1952ff2413b0f5ad4ad11f83f022f)
![{\displaystyle \int {\frac {dx}{\operatorname {ch} ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9bc39def5d5ff9c96e1f1234ee2d6dfa03859b)
- tudi:
![{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {1}{a}}\arctan(\operatorname {sh} ax)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa13a8afc8366d78d958757170e2af601bf231fa)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0088e7e6a327bf6f5161b71a6da1a52b6c03be6)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9596c789fa43a8396b809300315be29ec415a2a)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d1a8a7ebe99daf584d4daa92b6bd78a4c777fa)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/009bf0f38942492eecacf22ffb22193b71ff28cc)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff2ae0b46705b50d6b0180902ad537d1cb759e0)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61f99d51ee4d8fbcc2dc33a60e4dc20f5ffbfd1)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184c8345144d0a4b99b807add0da45122314e040)
- tudi:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aebadecfcc13cf31f356b50a1ead6d623215cb4a)
![{\displaystyle \int x\operatorname {sh} ax\,dx={\frac {1}{a}}x\operatorname {ch} ax-{\frac {1}{a^{2}}}\operatorname {sh} ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b94952e2bcd9d629299d019374e05c121649558)
![{\displaystyle \int x\operatorname {ch} ax\,dx={\frac {1}{a}}x\operatorname {sh} ax-{\frac {1}{a^{2}}}\operatorname {ch} ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aee5fc65aaaa190abd43a092bb56a0f688cb830)
![{\displaystyle \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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b949e55afa7611e087562b02d64061a4354df29)
![{\displaystyle \int \operatorname {th} ax\,dx={\frac {1}{a}}\ln \operatorname {ch} ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/996ea5b1deebc8e1d70aad056f5c566ed2dfddfe)
![{\displaystyle \int \operatorname {ch} ax\,dx={\frac {1}{a}}\ln |\operatorname {sh} ax|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675624f33bf367dfdce28894935e40163c8a280e)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29cff8979ea142edd96e89eb0f92e03c9f92f3d4)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/802b913cd18ddd50b4e814b38d5c7580e6806512)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e31ccdc952bd52791ff41fc3766a84b4ce9a210a)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28e2c43536aedf0c310815686dabfdb9995777a)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0ae6b14706d3c94f34a195474c69256af0d0daf)
![{\displaystyle \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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1784604867170df2c0e1a41610616f6e91ccdc73)
![{\displaystyle \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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/191977af9974f669001a389e36405ac9eb2772c1)
![{\displaystyle \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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbfeeca74217322aec2cd7d179dfe237159de29)
![{\displaystyle \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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81d31cca0f9c9eb05aaaba2c6694b1529d694cfa)