INTEGRALS CONTAINING $\displaystyle \frac{1}{\sinh ax}$

1.

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

2.

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

3.

$\displaystyle\int\displaystyle \frac{dx}{\sinh^3 ax}=-\displaystyle \frac{\coth ax}{2a\sinh ax}-\displaystyle \frac{1}{2a}\ln\tanh\displaystyle \frac{ax}{2}$

4.

$\displaystyle\int\displaystyle \frac{\coth ax}{\sinh^n ax}dx=-\displaystyle \frac{1}{na\sinh^n ax}$

5.

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

6.

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

where the constants B_n are the Bernoulli's numbers.

7.

$\displaystyle\int\displaystyle \frac{x dx}{\sinh^2 ax}=-\displaystyle \frac{x\coth ax}{a}+\displaystyle \frac{1}{a^2}\ln\sinh ax$

8.

$\displaystyle\int\displaystyle \frac{dx}{x\sinh ax}=-\displaystyle \frac{1}{ax}... ...aystyle \frac{(-1)^n 2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdot\cdot\cdot$

where the constants B_n are the Bernoulli's numbers.

9.

$\displaystyle\int\displaystyle \frac{dx}{q+p/\sinh ax}=\displaystyle \frac{x}{q}-\displaystyle \frac{p}{q}\int\displaystyle \frac{dx}{p+q/\sinh ax}$

10.

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