I would like to find the limit of $ \int_1^a \frac{\mathrm dt}{\sqrt{t(t-1)(a-t)}}$
when $ a\rightarrow1^+$ It seems that $ \int_1^a \frac{\mathrm dt}{\sqrt{t(t-1)(a-t)}}\sim_{a\rightarrow 1^+} \pi$
What bothers me is that $a$ is in the integrand and I cannot find an equivalent of $\frac{1}{\sqrt{t(t-1)(a-t)}}$ when $a\rightarrow1^+$
Moreover the integral $ \int \frac{\mathrm dt}{\sqrt{t(t-1)(a-t)}}$ "cannot be computed", is not simple.
Do you have any idea?