INTEGRALS CONTAINING Sinh(ax) and Cosh(ax)(

1.

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

2.

$$\int\sinh px\cosh qx dx=\displaystyle \frac{\cosh(p+q)x}{2(p+q)}+\displaystyle \frac{\cosh(p-q)x}{2(p-q)}$$

3.

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

4.

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

5.

$$\int\sinh^2 ax \cosh^2 ax dx=\displaystyle \frac{\sinh 4ax}{32a}-\displaystyle \frac{x}{8}$$

6.

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

7.

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

8.

$$\int\displaystyle \frac{dx}{\sinh ax\cosh^2 ax}=\displaystyle \frac{1}{a\cosh ax}+\displaystyle \frac{1}{a}\ln\tan\displaystyle \frac{ax}{2}$$

9.

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

10.

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

11.

$$\int\displaystyle \frac{\cosh^2 ax}{\sinh ax}dx=\displaystyle \frac{\cosh ax}{a}+\displaystyle \frac{1}{a}\ln\tan\displaystyle \frac{ax}{2}$$

12.

$$\int\displaystyle \frac{dx}{\cosh ax(1+\sinh ax)}=\displaystyle \f... ...yle \frac{1+\sinh ax}{\cosh ax}\right)+\displaystyle \frac{1}{a}\tan^{-1}e^{ax}$$

13.

$$\int\displaystyle \frac{dx}{\sinh ax(\cosh ax+1)}=\displaystyle \f... ...1}{2a}\ln\tanh\displaystyle \frac{ax}{2}+\displaystyle \frac{1}{2a(\cosh ax+1)}$$

14.

$$\int\displaystyle \frac{dx}{\sinh ax(\cosh ax-1)}=-\displaystyle \... ...1}{2a}\ln\tanh\displaystyle \frac{ax}{2}-\displaystyle \frac{1}{2a(\cosh ax-1)}$$

[Tables]

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