DEFINITE INTEGRALS CONTAINING HYPERBOLIC FUNCTIONS

1.

$$\int_{0}^{\infty}\displaystyle \frac{\sin ax}{\sinh bx}dx=\displaystyle \frac{\pi}{2b}\tanh\displaystyle \frac{a\pi}{2b}$$

2.

$$\int_{0}^{\infty}\displaystyle \frac{\cos ax}{\cosh bx}dx=\displaystyle \frac{\pi}{2b}\displaystyle \frac{1}{\cosh (a\pi/2b)}$$

3.

$$\int_{0}^{\infty}\displaystyle \frac{x dx}{\sinh ax}=\displaystyle \frac{\pi^2}{4a^2}$$

4.

$$\int_{0}^{\infty}\displaystyle \frac{x^n dx}{\sinh ax}=\displaysty... ...tyle \frac{1}{2^{n+1}}+\displaystyle \frac{1}{3^{n+1}}+\cdot\cdot\cdot \right\}$$

5.

$$\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}+1}dx=\displaystyle \frac{\pi}{2b}\csc\displaystyle \frac{a\pi}{b}-\displaystyle \frac{1}{2a}$$

6.

$$\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}-1}dx=\displaystyle \frac{1}{2a}-\displaystyle \frac{\pi}{2b}\cot\displaystyle \frac{a\pi}{b}$$

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