INTEGRALS CONTAINING Sinh(ax)

1.

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

2.

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

3.

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

4.

$\displaystyle\int\displaystyle \frac{\sinh ax}{x}dx=ax+\displaystyle \frac{(ax)^3}{3\cdot 3!}+\displaystyle \frac{(ax)^5}{5\cdot 5!}+\cdot\cdot\cdot$

5.

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

6.

$\displaystyle\int\displaystyle \frac{dx}{\sinh ax}=\displaystyle \frac{1}{a}\ln\tanh\displaystyle \frac{ax}{2}$

7.

$\displaystyle\int\displaystyle \frac{xdx}{\sinh ax}=\displaystyle \frac{1}{a^2}... ...tyle \frac{2(-1)^n(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}+\cdot\cdot\cdot \right\}$

8.

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

9.

$\displaystyle\int x\sinh^2 axdx=\displaystyle \frac{x\sinh 2ax}{4a}-\displaystyle \frac{\cosh 2ax}{8a^2}-\displaystyle \frac{x^2}{4}$

10.

$\displaystyle\int\displaystyle \frac{dx}{\sinh^2 ax}=-\displaystyle \frac{\coth ax}{a}$

11.

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

12.

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

13.

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

14.

$\displaystyle\int\displaystyle \frac{dx}{p+q\sinh ax}=\displaystyle \frac{1}{a\... ...+p-\displaystyle \sqrt{p^2+q^2}}{qe^{ax}+p+\displaystyle \sqrt{p^2+q^2}}\right)$

15.

$\displaystyle\int\displaystyle \frac{dx}{(p+q\sinh ax)^2}=\displaystyle \frac{-... ...nh ax)}+\displaystyle \frac{p}{p^2+q^2}\int\displaystyle \frac{dx}{p+q\sinh ax}$

16.

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

17.

$\displaystyle\int\displaystyle \frac{dx}{p^2-q^2\sinh^2 ax}=\displaystyle \frac... ...laystyle \sqrt{p^2+q^2}\tanh ax}{p-\displaystyle \sqrt{p^2+q^2}\tanh ax}\right)$

18.

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

19.

$\displaystyle\int\sinh^n axdx=\displaystyle \frac{\sinh^{n-1}ax\cosh ax}{an}-\displaystyle \frac{n-1}{n}\int\sinh^{n-2}axdx$

20.

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

21.

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

22.

$\displaystyle\int\displaystyle \frac{xdx}{\sinh^n ax}=\displaystyle \frac{-x\co... ...-2}ax}-\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{xdx}{\sinh^{n-2}ax}$