Numerical method to solve an integral that contains a singularity?

[amjad-sh]

Homework Statement

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Relevant Equations

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In fact I'm working on a condensed matter physics paper, where I stumbled with an integral that I need to visualize.
The function, Ls I need to visualize is equal to:
$$Ls=4\nu^4 \dfrac{\int_{-1}^{1} \dfrac{( 1-u^2)}{(u+\sqrt{u^2-\nu^2})^3} \, du}{\int_{-1}^{1}-u \Big (\dfrac{u-\sqrt{u^2-\nu^2}}{u+\sqrt{u^2+\nu^2}}\Big)^2 \, du}$$

I must draw this function interms of $\nu$, where $0\leq\nu \leq1$.
I tried to use Simpson's method by just taking the integral from zero to 1,but it gave me a totally different graph to what is depicted in the paper.
Is there a numerical method that can be used to approximate such integrals?

 

[jasonRF]

Why do you think integrals from 0 to 1 will be the same as integrals from -1 to 1?

Also, your title mentions a singularity. Could you tell us where you think the singularity is?

There is no general approach to numerically integrating a singular integrand. One approach is to simply evaluate the integral close to (but not at) the singularity; the challenge is that as you get closer and closer to the singularity you often need to use more points in your numerical integration rule to maintain accuracy. But sometimes a good first step is to see if cleverness can allow you to rewrite the integrand to allow you to extract the singular part out separately. In your case there is definitely a way to rewrite both of the integrals so that you don't actually need to numerically evaluate something that is singular.