Double exponential integration (a,∞) - how to implement.

[maistral]

NOTE: This isn't homework.

So I'm trying to integrate a really awkward integral with limits from a to infinity;

\int^{∞}_{30471.2729807}(\frac{83.1451 * 373.15}{X})-(\frac{83.1451 * 373.15}{X-30.4811353}-\frac{5534906.5380409}{X^2})dX

Since the Simpson's and Trapezoidal would be really awkward to use with these (I literally used a limit of 30471.2729807 to 1000000000 (lol)) I tried to search for other alternatives, and I found this.

I can't understand how this is implemented though. This transformation

\int^{∞}_{a}f(X)dX → \int^{∞}_{0}f(X+a)dX

is bothering me as well. Can someone point me where a step-by-step algorithm of the method's implementation is made; or if possible, someone tell me how it was done?

 

[UltrafastPED]

It's a transformation of variables: let u=X+a; then for X=(a,inf) we get u=(0,inf), and dX = du.

All they did was skip the formal analysis and went directly to the result: f(u) = f(X+a).

 

[maistral]

I see. Got it.

 

[SteamKing]

Instead of using trapezoidal or Simpson's rules to calculate the integral, Gaussian quadrature with the appropriate transformation of the semi-infinite interval might save you some time and be more accurate.

http://en.wikipedia.org/wiki/Gaussian_quadrature

http://en.wikipedia.org/wiki/Numerical_integration#Integrals_over_infinite_intervals