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?