Re: int(1/(1+x)^2/(1+k*x^(m+1)), x, 0, inf) with m>0 and k>0
- From: David Wilkinson <david@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 21 Oct 2005 07:29:42 +0100
JohnCreighton_@xxxxxxxxxxx wrote:
I tried MathCad and an early version of Maple and neither could integrate it. I am slightly rusty on the method of residues but doubt it could do this one. There are no singularities or poles in the range of integration for k,m > 0. There would be one at x = -1 but this probably does not help as it is outside the range of integration.xinyu.xia@xxxxxxxxx wrote:
Could anyone help me to solve the following integral?
int(1/(1+x)^2/(1+k*x^(m+1)), x, 0, inf) with m>0 and k>0
I have tried with MATLAB 7.0 several times but it gave no solution!
I think you can do it by residues and maybe also by partial fractions depending on what m is. It has been a long time since I solved an integral using residues but pick up a book on complex variables.
There are no exact integrals for a lot of functions but they can generally be integrated numerically. The integrand looks remarkably simple in behaviour, starting at 1.0 for x = 0 and declining rapidly towards zero as x increases. For instance, for k=m=1, the integrand has dropped to 1/8 by x = 1 and to 1/45 at x = 2.
If you want an answer to any specified accuracy then the tricky bit is that the integral is "improper" as its upper limit is at infinity. A way round this is to split the range into two integrals.
The first integral is, say, from x = 0 to 1 which can be found numerically to any accuracy as it stands.
The second integral is from x = 1 to infinity and here you can probably use the transformation int(f(x),x=a..b) = int(1/t^2*f(1/t),t=1/a..1/b), (Numerical Recipes, Chapter 4, Section 4.4).
Since a = 1 and b = Inf, the range is now from t = 1 to 0 or 0 to 1. This works for any function that tends to zero at infinity at least as fast as 1/x^2, which this one does. Since the new integrand and the range are now finite this second integral can also be found numerically to any accuracy. Of course, it is possible that Mat lab or Maple will do all this for you if you go for a numerical integration with an upper limit of infinity.
Hope this helps. .
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