INTEGRALS CONTAINING Coth(ax)

1.

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

2.

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

3.

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

4.

$\displaystyle\int\displaystyle \frac{\coth^n ax}{\sinh^2 ax}dx=-\displaystyle \frac{\coth^{n+1}ax}{(n+1)a}$

5.

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

6.

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

7.

$\displaystyle\int x\coth axdx=\displaystyle \frac{1}{a^2}\left\{ax+\displaystyl... ...ystyle \frac{(-1)^{n-1}2^{2n}B_n(ax)^{2n+1}}{(2n+1)!}+ \cdot\cdot\cdot \right\}$

where the constants B_n are the Bernoulli's numbers.

8.

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

9.

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

where the constants B_n are the Bernoulli's numbers.

10.

$\displaystyle\int\displaystyle \frac{dx}{p+q\coth ax}=\displaystyle \frac{px}{p^2-q^2}-\displaystyle \frac{q}{a(p^2-q^2)}\ln(p\sinh ax+q\cosh ax)$

11.

$\displaystyle\int\coth^n ax dx=-\displaystyle \frac{\coth^{n-1}ax}{a(n-1)}+\int\coth^{n-1}axdx$