Վիքիպեդիայից՝ ազատ հանրագիտարանից
Այց ցանկում ընդգրկված են հիպերբոլական ֆունկցիաների ինտեգրալները։ Անորոշ ինտեգրալների համար ինտեգրման հաստատունը բաց է թողնված։
![{\displaystyle \int \operatorname {sh} cx\,dx={\frac {1}{c}}\operatorname {ch} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab2594333dcdc3ca27c9772ed5e8e2bb16d2e0d)
![{\displaystyle \int \operatorname {ch} cx\,dx={\frac {1}{c}}\operatorname {sh} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a186846e7a2459528f1beb55d105994bb95b3a9)
![{\displaystyle \int \operatorname {sh} ^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh} 2cx-{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd66b13a4b6717056ade03a500d2a6fc7dc3a4a7)
![{\displaystyle \int \operatorname {ch} ^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh} 2cx+{\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8b755eeda5c4ab2758f774f596c7510c4bfee2)
![{\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{cn}}\operatorname {sh} ^{n-1}cx\operatorname {ch} cx-{\frac {n-1}{n}}\int \operatorname {sh} ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d734ac684837564bdde0b724b1c47c30eb3a65)
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![{\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{c(n+1)}}\operatorname {sh} ^{n+1}cx\operatorname {ch} cx-{\frac {n+2}{n+1}}\int \operatorname {sh} ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f69b921d72d5fbc148f325d7756b20888e1c5003)
![{\displaystyle \int \operatorname {ch} ^{n}cx\,dx={\frac {1}{cn}}\operatorname {sh} cx\operatorname {ch} ^{n-1}cx+{\frac {n-1}{n}}\int \operatorname {ch} ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13f48a5b8eb001e24963bbe7358c894aa02cab4e)
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![{\displaystyle \int \operatorname {ch} ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\operatorname {sh} cx\operatorname {ch} ^{n+1}cx-{\frac {n+2}{n+1}}\int \operatorname {ch} ^{n+2}cx\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd48e16d0489cb67fed505ed3e325616c520ffb)
![{\displaystyle \int {\frac {dx}{\operatorname {sh} cx}}={\frac {1}{c}}\ln \left|\operatorname {th} {\frac {cx}{2}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {sh} cx}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {sh} cx}{\operatorname {ch} cx+1}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {ch} cx+1}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb172f07cc5a8f8d82a7eacd3dd62d72d4cf38d)
![{\displaystyle \int {\frac {dx}{\operatorname {sh} ^{2}cx}}=-{\frac {1}{c}}\operatorname {cth} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d1626845a1342b860c8494787d3fadb3fb40de)
![{\displaystyle \int {\frac {dx}{\operatorname {ch} cx}}={\frac {2}{c}}\operatorname {arctg} e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9b0b2af57cef006368d180457de221b7b66328)
![{\displaystyle \int {\frac {dx}{\operatorname {ch} ^{2}cx}}={\frac {1}{c}}\operatorname {th} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d095a3b963104410e8baa3398c8aaf44980778a)
![{\displaystyle \int {\frac {dx}{\operatorname {sh} ^{n}cx}}={\frac {\operatorname {ch} cx}{c(n-1)\operatorname {sh} ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\operatorname {sh} ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839355f51450d2667c92c3bcb621098aaf517e3b)
![{\displaystyle \int {\frac {dx}{\operatorname {ch} ^{n}cx}}={\frac {\operatorname {sh} cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\operatorname {ch} ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0010adc5a04c351f3f2995e1462ff5bb23bfcd23)
![{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx={\frac {\operatorname {ch} ^{n-1}cx}{c(n-m)\operatorname {sh} ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\operatorname {ch} ^{n-2}cx}{\operatorname {sh} ^{m}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c83a90d6724d8d9472229d53ec01c89b10d59fc)
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![{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx=-{\frac {\operatorname {ch} ^{n+1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a1c679971f5c708d5fe007200dc695a83d17d3)
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![{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx=-{\frac {\operatorname {ch} ^{n-1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\operatorname {ch} ^{n-2}cx}{\operatorname {sh} ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/759cb9e43490001e30fdc33b87374ecc20c2dbdd)
![{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx={\frac {\operatorname {sh} ^{m-1}cx}{c(m-n)\operatorname {ch} ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\operatorname {sh} ^{m-2}cx}{\operatorname {ch} ^{n}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c62b14a9fcfcc9fcbe26af7e386eb93d03875bde)
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![{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx={\frac {\operatorname {sh} ^{m+1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00a3972ee5e6717fea1581cbe3b15364d730f8a8)
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![{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx=-{\frac {\operatorname {sh} ^{m-1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\operatorname {sh} ^{m-2}cx}{\operatorname {ch} ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2622912dc8947fd6f484b33bf9292f06e4b2c5e6)
![{\displaystyle \int x\operatorname {sh} cx\,dx={\frac {1}{c}}x\operatorname {ch} cx-{\frac {1}{c^{2}}}\operatorname {sh} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9412c25e2738e3238110b4efc931c60263579dbe)
![{\displaystyle \int x\operatorname {ch} cx\,dx={\frac {1}{c}}x\operatorname {sh} cx-{\frac {1}{c^{2}}}\operatorname {ch} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d4763674e8265daabe1a39b7976816e72437d)
![{\displaystyle \int \operatorname {th} cx\,dx={\frac {1}{c}}\ln |\operatorname {ch} cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b325535e1e6570f396585f86ce50db349e1db7e)
![{\displaystyle \int \operatorname {cth} cx\,dx={\frac {1}{c}}\ln |\operatorname {sh} cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4cbf52f191745dab9e5c65b05f8f6450cc353f)
![{\displaystyle \int \operatorname {th} ^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {th} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94d0a68630aea8a895a1fa05d7a9af18470fca90)
![{\displaystyle \int \operatorname {cth} ^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {cth} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5da7cf4580a34d7e514153a079d376a580d5f701)
![{\displaystyle \int \operatorname {th} ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\operatorname {th} ^{n-1}cx+\int \operatorname {th} ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{ )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09cbca78121a966bf99e3f30b55fd8a86431163e)
![{\displaystyle \int \operatorname {cth} ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\operatorname {cth} ^{n-1}cx+\int \operatorname {cth} ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0657b0a851930914aad215d16e9dae4256119b3b)
![{\displaystyle \int \operatorname {sh} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} cx\operatorname {ch} bx-c\operatorname {ch} cx\operatorname {sh} bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1868107e74a7ee06cf1783c9095a509574133c1)
![{\displaystyle \int \operatorname {ch} bx\operatorname {ch} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} bx\operatorname {ch} cx-c\operatorname {sh} cx\operatorname {ch} bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6190ad3f98e2cb54af9cd3a0358f962da4012a3)
![{\displaystyle \int \operatorname {ch} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} bx\operatorname {sh} cx-c\operatorname {ch} bx\operatorname {ch} cx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ebe5c99c61295cd829b18c9998852e827b0bb)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43bbb9cc0be53dd341c62c835ae5ee383cbb98ff)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e59b345546da107eb94f968f58b661e7f72545)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56f3238c9e0f194953289b9f15e649d61d180756)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0594175f06be7e5a1808e28dee813612a7d0e16a)
- Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений (4-е издание). М.։ Наука, 1963. ISBN 0-12-294757-6 // EqWorld
- Двайт Г. Б. Таблицы интегралов СПб։ «Издательство и типография АО ВНИИГ им. Б. В. Веденеева», 1995.-176 с. ISBN 5-85529-029-8.
- D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4