Список на интеграли на хиперболични функции

Списoк na интеграли на хиперболични функции:

$\int \sinh cxdx=\frac{1}{c}\cosh cx$

$\int \cosh cxdx=\frac{1}{c}\sinh cx$

$\int {\sinh }^{2}cxdx=\frac{1}{4c}\sinh 2cx-\frac{x}{2}$

$\int {\cosh }^{2}cxdx=\frac{1}{4c}\sinh 2cx+\frac{x}{2}$

$\int {\sinh }^{n}cxdx=\frac{1}{cn}{\sinh }^{n-1}cx\cosh cx-\frac{n-1}{n}\int {\sinh }^{n-2}cxdx\text{(for }n>0\text{)}$

исто така: $\int {\sinh }^{n}cxdx=\frac{1}{c(n+1)}{\sinh }^{n+1}cx\cosh cx-\frac{n+2}{n+1}\int {\sinh }^{n+2}cxdx\text{(for }n<0\text{, }n\ne -1\text{)}$

$\int {\cosh }^{n}cxdx=\frac{1}{cn}\sinh cx{\cosh }^{n-1}cx+\frac{n-1}{n}\int {\cosh }^{n-2}cxdx\text{(for }n>0\text{)}$

исто така: $\int {\cosh }^{n}cxdx=-\frac{1}{c(n+1)}\sinh cx{\cosh }^{n+1}cx-\frac{n+2}{n+1}\int {\cosh }^{n+2}cxdx\text{(for }n<0\text{, }n\ne -1\text{)}$

$\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln |\tanh \frac{cx}{2}|$

исто така: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\sinh cx}\right|$

исто така: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\sinh cx}{\cosh cx+1}\right|$

исто така: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\cosh cx+1}\right|$

$\int \frac{dx}{\cosh cx}=\frac{2}{c}\arctan e^{cx}$

$\int \frac{dx}{{\sinh }^{n}cx}=\frac{\cosh cx}{c(n-1){\sinh }^{n-1}cx}-\frac{n-2}{n-1}\int \frac{dx}{{\sinh }^{n-2}cx}\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{{\cosh }^{n}cx}=\frac{\sinh cx}{c(n-1){\cosh }^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\cosh }^{n-2}cx}\text{(for }n\ne 1\text{)}$

$\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=\frac{{\cosh }^{n-1}cx}{c(n-m){\sinh }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\cosh }^{n-2}cx}{{\sinh }^{m}cx}dx\text{(for }m\ne n\text{)}$

исто така: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n+1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-m+2}{m-1}\int \frac{{\cosh }^{n}cx}{{\sinh }^{m-2}cx}dx\text{(for }m\ne 1\text{)}$

исто така: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n-1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-1}{m-1}\int \frac{{\cosh }^{n-2}cx}{{\sinh }^{m-2}cx}dx\text{(for }m\ne 1\text{)}$

$\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=\frac{{\sinh }^{m-1}cx}{c(m-n){\cosh }^{n-1}cx}+\frac{m-1}{m-n}\int \frac{{\sinh }^{m-2}cx}{{\cosh }^{n}cx}dx\text{(for }m\ne n\text{)}$

исто така: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=\frac{{\sinh }^{m+1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-n+2}{n-1}\int \frac{{\sinh }^{m}cx}{{\cosh }^{n-2}cx}dx\text{(for }n\ne 1\text{)}$

исто така: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=-\frac{{\sinh }^{m-1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-1}{n-1}\int \frac{{\sinh }^{m-2}cx}{{\cosh }^{n-2}cx}dx\text{(for }n\ne 1\text{)}$

$\int x\sinh cxdx=\frac{1}{c}x\cosh cx-\frac{1}{c^2}\sinh cx$

$\int x\cosh cxdx=\frac{1}{c}x\sinh cx-\frac{1}{c^2}\cosh cx$

$\int \tanh cxdx=\frac{1}{c}\ln |\cosh cx|$

$\int \coth cxdx=\frac{1}{c}\ln |\sinh cx|$

$\int {\tanh }^{n}cxdx=-\frac{1}{c(n-1)}{\tanh }^{n-1}cx+\int {\tanh }^{n-2}cxdx\text{(for }n\ne 1\text{)}$

$\int {\coth }^{n}cxdx=-\frac{1}{c(n-1)}{\coth }^{n-1}cx+\int {\coth }^{n-2}cxdx\text{(for }n\ne 1\text{)}$

$\int \sinh bx\sinh cxdx=\frac{1}{b^2-c^2}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\text{(for }{b}^2\ne c^2\text{)}$

$\int \cosh bx\cosh cxdx=\frac{1}{b^2-c^2}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\text{(for }{b}^2\ne c^2\text{)}$

$\int \cosh bx\sinh cxdx=\frac{1}{b^2-c^2}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\text{(for }{b}^2\ne c^2\text{)}$

$\int \sinh (ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\cosh (ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\sinh (ax+b)\cos (cx+d)$

$\int \sinh (ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\cosh (ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\sinh (ax+b)\sin (cx+d)$

$\int \cosh (ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\sinh (ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\cosh (ax+b)\cos (cx+d)$

$\int \cosh (ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\sinh (ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\cosh (ax+b)\sin (cx+d)$

Литература

• Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
• I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)
• Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
• Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)