# Double exponential integration (a,b) Calculator

## Calculates the integral of the given function f(x) over the interval (a,b) using Double exponential formula.

 This method is suitable for the function with endpoint singularities (±∞). The integrand f(x) is assumed to be analytic and non-periodic.
 f(x) a , b
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Double\ exponential\ integration\\(1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\\hspace{30} {\large\int_{\small a}^{\hspace{25}\small b}}f(x)dx= {\large\int_{\small -1}^{\hspace{25}\small 1}}f(\frac{b-a}{2}x+\frac{b+a}{2})\frac{b-a}{2}dx\\[10](2) x\rightarrow tanh(\frac{\pi}{2}sinh(t))\\\hspace{30}{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx={\large\int_{\small -\infty}^{\hspace{25}\small \infty}}f(x(t))x'(t)dt\\\hspace{140}x(t)=tanh(\frac{\pi}{2}sinh(t))\\\hspace{140}x'(t)={\large\frac{\frac{\pi}{2}cosh(t)}{cosh^2(\frac{\pi}{2}sinh(t))}}\\[10](3)\ Trapezoid\\\hspace{20}S\simeq{\large\int_{-t_a}^{\hspace{25}t_a}}f(x(t))x'(t)dt=h{\large\sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\\hspace{240}h={\large\frac{2t_a}{N}}\\$

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