# Numerical method to solve an integral that contains a singularity?

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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?

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