# Double exponential integration (a,∞) Calculator

## Calculates the integral of the given function f(x) over the semi-infinite interval (a,∞) 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)
a

integral S   error

 $\normal Double\ exponential\ integration\\(1)\ x\rightarrow e^{{\normal\frac{\pi}{2}}sinh(t)}\\\ {\large\int_{\tiny a}^{\hspace{25}\tiny\infty}}f(x)dx={\large\int_{\tiny 0}^{\hspace{25}\tiny\infty}}f(x+a)dx={\large\int_{\tiny -\infty}^{\hspace{25}\tiny\infty}}f(x(t)+a)x'(t)dt\\\hspace{140}x(t)=e^{{\normal\frac{\pi}{2}}sinh(t)}\\\hspace{140}x'(t)=e^{{\normal\frac{\pi}{2}}sinh(t)}\frac{\pi}{2}cosh(t)\\[10](2)\ Trapezoid\\\hspace{20}S\simeq{\large\int_{-t_a}^{\hspace{25}t_a}}f(x(t))x'(t)dt=h{\large\sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\\hspace{240}h={\large\frac{2t_a}{N}}\\$

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