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?