Vikipedi, özgür ansiklopedi
Aşağıdaki liste hiperbolik fonksiyonların integrallerini içermektedir. İntegral fonksiyonlarının tüm bir listesi için lütfen İntegral tablosu sayfasına bakınız.
c sabiti sıfırdan farklı varsayılmıştır.
![{\displaystyle \int \sinh cx\,dx={\frac {1}{c}}\cosh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c4b91d3104f80c73901b809ba4eb99f54b810a)
![{\displaystyle \int \cosh cx\,dx={\frac {1}{c}}\sinh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/567ad24b724ce699659c1f2fe458873890caacfb)
![{\displaystyle \int \sinh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1c16e730629bd2826a542e2badcb6c6170156f)
![{\displaystyle \int \cosh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17cb560251e82ebce9a8e483553268b6bba8f5b9)
![{\displaystyle \int \tanh ^{2}cx\,dx=x-{\frac {\tanh cx}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6ce2030e7258d65468ba767ee9eadf87198f17)
![{\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68931a24b7f70cad57308f1bf71dc93e7ebc7d13)
- Ayrıca:
![{\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3963fae692212263f85d2565f02be03b5effb06e)
![{\displaystyle \int \cosh ^{n}cx\,dx={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/371443bd2fe33e4b188c8b6194759f667b0718a7)
- Ayrıca:
![{\displaystyle \int \cosh ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb0c65c2e95b6a1b6ae30def7242799119a69f2)
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2c2e54d71d6f636aa16a52f14ad2acca1905a9)
- Ayrıca:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b1d39870d3500e34a07fef57ea341eec9ae0e3)
- Ayrıca:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f99aba69af941713d3f013c5ed15062761c0c2)
- Ayrıca:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc369affe8fa0f14bff643a02c7a70df684d6f35)
![{\displaystyle \int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b31994778e3068c923c293c7d283b631620616)
![{\displaystyle \int {\frac {dx}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5900e971eeadeb1f832cd5b9c54a5137eb4c2589)
![{\displaystyle \int {\frac {dx}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdb8c8b809a3b711e885b931feb3b792aae14ca)
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9530dfae977cf76936cb10d2085ddfb834d9e3f3)
- Ayrıca:
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5376f51bb11bda00e8c5456b7613e44c4c9a7b84)
- Ayrıca:
![{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ecd4185e3bec010de7535a1c2e8d39d15473e1)
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8203e9aa67c1e96c2d3e9dfa5434eec595a2859a)
- Ayrıca:
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9445e8a6fcdcc4bb674d7648cf3e133cacc265b2)
- Ayrıca:
![{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb63803f332dc8fc75e53efaccb4b4d3fab609e8)
![{\displaystyle \int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e47e20391f6a47ab1157165e25bda8464755661)
![{\displaystyle \int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e24c60a1200c219281a47ef75bd9da046754afb)
![{\displaystyle \int x^{2}\cosh cx\,dx=-{\frac {2x\cosh cx}{c^{2}}}+\left({\frac {x^{2}}{c}}+{\frac {2}{c^{3}}}\right)\sinh cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/136f10b362a465cefe0dc725b5d38eddcae2777d)
![{\displaystyle \int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e4afe23edbd31d87d1bce506d3181c5799fe9f)
![{\displaystyle \int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69406501255bff4de4f4f0e3a8ddd55b96727b7f)
![{\displaystyle \int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d86d363c43850e3061c8e40ab29d044fd56488)
![{\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34851b9a072f819c1ec1198e1a3b4c17882f79cb)
![{\displaystyle \int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(for }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52f206e8628a22500673730bdd3b3a7c507f9d41)
![{\displaystyle \int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(for }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a19d64c86a087866200364dc3667e0ef7917778)
![{\displaystyle \int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(for }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2053b7741240ea1be6ba8931262312a74b07ad)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a5279ec5fbc713e90ac86b42a8e8dcc2909626)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f264b7005d83c940ba683472e1c933be9f2d9ca4)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16c98a2c6dea24a8ef6409d046c9a83c03b3e637)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca54f3708fa0fa6500dd9bbf81e0064f808e4291)