Списак интеграла хиперболичких функција

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Списак интеграла хиперболичких функција:

\int\sinh cx\,dx = \frac{1}{c}\cosh cx
\int\cosh cx\,dx = \frac{1}{c}\sinh cx
\int\sinh^2 cx\,dx = \frac{1}{4c}\sinh 2cx - \frac{x}{2}
\int\cosh^2 cx\,dx = \frac{1}{4c}\sinh 2cx + \frac{x}{2}
\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{)}
такође: \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{)}
\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{)}
такође: \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{)}
\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\tanh\frac{cx}{2}\right|
такође: \int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\sinh cx}\right|
такође: \int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\sinh cx}{\cosh cx + 1}\right|
такође: \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} \qquad\mbox{(for }n\neq 1\mbox{)}
\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{)}
\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{)}
такође: \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{)}
такође: \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{)}
\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{)}
такође: \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{)}
такође: \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{)}
\int x\sinh cx\,dx = \frac{1}{c} x\cosh cx - \frac{1}{c^2}\sinh cx
\int x\cosh cx\,dx = \frac{1}{c} x\sinh cx - \frac{1}{c^2}\cosh cx
\int \tanh cx\,dx = \frac{1}{c}\ln|\cosh cx|
\int \coth cx\,dx = \frac{1}{c}\ln|\sinh cx|
\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{)}
\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{)}
\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{)}
\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{)}
\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{)}
\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[уреди]

  • 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)