This question is similar to my previous one:
I would like to find the limit of $ \int_0^a \sqrt{\frac{x^2+1}{x(a-x)}} \mathrm dx$
when $ a\rightarrow 0^+$ Once again it seems that $ \int_0^a \sqrt{\frac{x^2+1}{x(a-x)}} \mathrm dx\sim_{a\rightarrow 1^+} \pi$
We have:
$ \sqrt{\frac{x^2+1}{x(a-x)}}=\frac{2}{a}\sqrt{\frac{x^2+1}{1-(\frac{2x}{a}-1)^2}} $
Does this help find a suitable change of variable?