INTEGRALS CONTAINING Cosh(ax)

1.
$\displaystyle\int\cosh axdx=\displaystyle \frac{\sinh ax}{a}$

2.
$\displaystyle\int x\cosh axdx=\displaystyle \frac{x\sinh ax}{a}-\displaystyle \frac{\cosh ax}{a^2}$

3.
$\displaystyle\int x^2\cosh axdx=-\displaystyle \frac{2x\cosh ax}{a^2}+\left( \displaystyle \frac{x^2}{a}+\displaystyle \frac{2}{a^3}\right) \sinh ax$

4.
$\displaystyle\int\displaystyle \frac{\cosh ax}{x}dx=\ln x+\displaystyle \frac{(...
...\frac{(ax)^4}{4\cdot 4!}+\displaystyle \frac{(ax)^6}{6\cdot 6!}+\cdot\cdot\cdot$

5.
$\displaystyle\int\displaystyle \frac{\cosh ax}{x^2}dx=-\displaystyle \frac{\cosh ax}{x}+a\int\displaystyle \frac{\sinh ax}{x}dx$

6.
$\displaystyle\int\displaystyle \frac{dx}{\cosh ax}=\displaystyle \frac{2}{a}\tan^{-1e^{ax}}$

7.
$\displaystyle\int\displaystyle \frac{x dx}{\cosh ax}=\displaystyle \frac{1}{a^2...
...playstyle \frac{(-1)^n E_{n}(ax)^{2n+2}}{(2n+2)(2n)!}+ \cdot\cdot\cdot \right\}$

where the constants En are the Euler's numbers.

8.
$\displaystyle\int\cosh^2 ax dx=\displaystyle \frac{x}{2}+\displaystyle \frac{\sinh ax \cosh ax}{2}$

9.
$\displaystyle\int x\cosh^2 ax dx=\displaystyle \frac{x^2}{4}+\displaystyle \frac{x\sinh 2ax}{4a}-\displaystyle \frac{\cosh 2ax}{8a^2}$

10.
$\displaystyle\int\displaystyle \frac{dx}{\cosh^2 ax}=\displaystyle \frac{\tanh ax}{a}$

11.
$\displaystyle\int\cosh ax\cosh px dx=\displaystyle \frac{\sinh(a-p)x}{2(a-p)}+\displaystyle \frac{\sinh(a+p)2}{2(a+p)}$

12.
$\displaystyle\int\cosh ax\sin px dx=\displaystyle \frac{a\sinh ax\sin px -p\cosh ax\cos px}{a^2+p^2}$

13.
$\displaystyle\int\cosh ax \cos pxdx=\displaystyle \frac{a\sinh ax\cos px+p\cosh ax\sin px}{a^2+p^2}$

14.
$\displaystyle\int\displaystyle \frac{dx}{\cosh ax+1}=\displaystyle \frac{1}{a}\tanh\displaystyle \frac{ax}{2}$

15.
$\displaystyle\int\displaystyle \frac{dx}{\cosh ax-1}=-\displaystyle \frac{1}{a}\coth\displaystyle \frac{ax}{2}$

16.
$\displaystyle\int\displaystyle \frac{x dx}{\cosh ax+1}=\displaystyle \frac{x}{a...
...tyle \frac{ax}{2}-\displaystyle \frac{2}{a^2}\ln\cosh\displaystyle \frac{ax}{2}$

17.
$\displaystyle\int\displaystyle \frac{x dx}{\cosh ax-1}=-\displaystyle \frac{x}{...
...tyle \frac{ax}{2}+\displaystyle \frac{2}{a^2}\ln\sinh\displaystyle \frac{ax}{2}$

18.
$\displaystyle\int\displaystyle \frac{dx}{(\cosh ax+1)^2}=\displaystyle \frac{1}...
...ystyle \frac{ax}{2}-\displaystyle \frac{1}{6a}\tanh^3\displaystyle \frac{ax}{2}$

19.
$\displaystyle\int\displaystyle \frac{dx}{(\cosh ax-1)^2}=\displaystyle \frac{1}...
...ystyle \frac{ax}{2}-\displaystyle \frac{1}{6a}\coth^3\displaystyle \frac{ax}{2}$

20.
$\displaystyle\int\displaystyle \frac{dx}{p+q\cosh ax}=\left\{ \begin{array}{ll}...
...rt{p^2-q^2}}{qe^{ax}+p+\displaystyle \sqrt{p^2-q^2}}\right)
\end{array}\right. $

21.
$\displaystyle\int\displaystyle \frac{dx}{(p+q\cosh ax)^2}=\displaystyle \frac{q...
...sh ax)}-\displaystyle \frac{p}{q^2-p^2}\int\displaystyle \frac{dx}{p+q\cosh ax}$

22.
$\displaystyle\int\displaystyle \frac{dx}{p^2-q^2\cosh^2 ax}=\left\{ \begin{arra...
...isplaystyle \frac{p\tanh ax}{\displaystyle \sqrt{q^2-p^2}}
\end{array}\right. $

23.
$\displaystyle\int\displaystyle \frac{dx}{p^2+q^2\cosh^2 ax}=\left\{ \begin{arra...
...displaystyle \frac{p\tanh ax}{\displaystyle \sqrt{p^2+q^2}}
\end{array}\right. $

24.
$\displaystyle\int x^m \cosh ax dx=\displaystyle \frac{x^m \sinh ax}{a}-\displaystyle \frac{m}{a}\int x^{m-1}\sinh ax dx$

25.
$\displaystyle\int\cosh^n ax dx=\displaystyle \frac{\cosh^{n-1}ax\sinh ax}{an}+\displaystyle \frac{n-1}{n}\int\cosh^{n-2} ax dx$

26.
$\displaystyle\int\displaystyle \frac{\cosh ax}{x^n}dx=\displaystyle \frac{-\cos...
...^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{\sinh ax}{x^{n-1}}dx$

27.
$\displaystyle\int\displaystyle \frac{dx}{\cosh^n ax}=\displaystyle \frac{\sinh ...
...n-1}ax}+\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{dx}{\cosh^{n-2}ax}$

28.
$\displaystyle\int\displaystyle \frac{x dx}{\cosh^n ax}=\displaystyle \frac{x\si...
...2}ax}+\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{x dx}{\cosh^{n-2}ax}$

[Tables]

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