Question: Help! How to numerically integrate a function which has a singularity and a very sharp peak?

Hi all,

Here is a description to the gamma density function:

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

Basically, I am studying how to numerically integrate against the
gamma density function for kappa < 1.

As you can see, the key difficulty for the kappa < 1 case is that in
the integrand there is a term

x^(kappa-1)

and we have to integrate starting from x=0 all the way to x= +
infinity...

There is a very sharp peak at x=0, which leads to a lot of difficulty
for my Simpson's rule numerical integration scheme.

For some reason, I have to integrate by calculating the nodes and
weights myself. This is actually good - it helps me understand the
numerical integration better.

I want to learn the tricks in numerical integration field about how
experts handle such case...

Please shed some light on me...

Thanks a lot for your help!