INTEGRALS CONTAINING Cosh(ax)

1.

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

2.

$$\int x\cosh axdx=\displaystyle \frac{x\sinh ax}{a}-\displaystyle \frac{\cosh ax}{a^2}$$

3.

$$\int x^2\cosh axdx=-\displaystyle \frac{2x\cosh ax}{a^2}+\left( \displaystyle \frac{x^2}{a}+\displaystyle \frac{2}{a^3}\right) \sinh ax$$

4.

$$\int\displaystyle \frac{\cosh ax}{x}dx=\ln x+\displaystyle \frac{(... ...\frac{(ax)^4}{4\cdot 4!}+\displaystyle \frac{(ax)^6}{6\cdot 6!}+\cdot\cdot\cdot$$

5.

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

6.

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

7.

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

where the constants E_n are the Euler's numbers.

8.

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

9.

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

10.

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

11.

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

12.

$$\int\cosh ax\sin px dx=\displaystyle \frac{a\sinh ax\sin px -p\cosh ax\cos px}{a^2+p^2}$$

13.

$$\int\cosh ax \cos pxdx=\displaystyle \frac{a\sinh ax\cos px+p\cosh ax\sin px}{a^2+p^2}$$

14.

$$\int\displaystyle \frac{dx}{\cosh ax+1}=\displaystyle \frac{1}{a}\tanh\displaystyle \frac{ax}{2}$$

15.

$$\int\displaystyle \frac{dx}{\cosh ax-1}=-\displaystyle \frac{1}{a}\coth\displaystyle \frac{ax}{2}$$

16.

$$\int\displaystyle \frac{x dx}{\cosh ax+1}=\displaystyle \frac{x}{a... ...tyle \frac{ax}{2}-\displaystyle \frac{2}{a^2}\ln\cosh\displaystyle \frac{ax}{2}$$

17.

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

18.

$$\int\displaystyle \frac{dx}{(\cosh ax+1)^2}=\displaystyle \frac{1}... ...ystyle \frac{ax}{2}-\displaystyle \frac{1}{6a}\tanh^3\displaystyle \frac{ax}{2}$$

19.

$$\int\displaystyle \frac{dx}{(\cosh ax-1)^2}=\displaystyle \frac{1}... ...ystyle \frac{ax}{2}-\displaystyle \frac{1}{6a}\coth^3\displaystyle \frac{ax}{2}$$

20.

$$\int\displaystyle \frac{dx}{p+q\cosh ax}=\left\{ \begin{array}{ll}... ...rt{p^2-q^2}}{qe^{ax}+p+\displaystyle \sqrt{p^2-q^2}}\right) \end{array}\right.$$

21.

$$\int\displaystyle \frac{dx}{(p+q\cosh ax)^2}=\displaystyle \frac{q... ...sh ax)}-\displaystyle \frac{p}{q^2-p^2}\int\displaystyle \frac{dx}{p+q\cosh ax}$$

22.

$$\int\displaystyle \frac{dx}{p^2-q^2\cosh^2 ax}=\left\{ \begin{arra... ...isplaystyle \frac{p\tanh ax}{\displaystyle \sqrt{q^2-p^2}} \end{array}\right.$$

23.

$$\int\displaystyle \frac{dx}{p^2+q^2\cosh^2 ax}=\left\{ \begin{arra... ...displaystyle \frac{p\tanh ax}{\displaystyle \sqrt{p^2+q^2}} \end{array}\right.$$

24.

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

25.

$$\int\cosh^n ax dx=\displaystyle \frac{\cosh^{n-1}ax\sinh ax}{an}+\displaystyle \frac{n-1}{n}\int\cosh^{n-2} ax dx$$

26.

$$\int\displaystyle \frac{\cosh ax}{x^n}dx=\displaystyle \frac{-\cos... ...^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{\sinh ax}{x^{n-1}}dx$$

27.

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

28.

$$\int\displaystyle \frac{x dx}{\cosh^n ax}=\displaystyle \frac{x\si... ...2}ax}+\displaystyle \frac{n-2}{n-1}\int\displaystyle \frac{x dx}{\cosh^{n-2}ax}$$

[Tables]

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