# Double exponential integration (-∞,∞)

## Calculates the integral of the given function f(x) over the infinite interval (-∞,∞) using the Double exponential formula.

 This method is suitable for the function with algebraic decay at infinity.The integrand f(x) is assumed to be analytic and non-periodic except for the endpoint.
 f(x)
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Double\ exponential\ integration\\[10](1)\ x\rightarrow sinh(\frac{\pi}{2}sinh(t))\\[10]\hspace{10} {\large\int_{\small -\infty}^{\hspace{25}\small\infty}}f(x)dx={\large\int_{\small -\infty}^{\hspace{25}\small\infty}}f(x(t))x'(t)dt\\\hspace{100}x(t)=sinh(\frac{\pi}{2}sinh(t))\\\hspace{100}x'(t)=cosh(\frac{\pi}{2}sinh(t))\frac{\pi}{2}cosh(t)\\[10](2)\ Trapezoid\\\hspace{20}S\simeq{\large\int_{-t_a}^{\hspace{25}t_a}}f(x(t))x'(t)dt=h{\large\sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\\hspace{240}h={\large\frac{2t_a}{N}}\\$

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