INTEGRALS CONTAINING Coth(ax)

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

where the constants B_n are the Bernoulli's numbers.

8.

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

9.

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

where the constants B_n are the Bernoulli's numbers.

10.

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

11.

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

[Tables]

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