Computing $\int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}}$ 
$$
\int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}}
$$
how? $x^2/\sqrt{8x-x^2-15}$ and what to do then? 

Set $x=3+2\sin^2(t)$. We then obtain
$$\int_3^5 \dfrac{x^2dx}{\sqrt{(x-3)(5-x)}} = \int_0^{\pi/2} \dfrac{2(3+2\sin^2(t))\sin(2t)dt}{\sqrt{(2\sin^2(t))\cdot(2\cos^2(t))}} = \int_0^{\pi/2}2(3+2\sin^2(t))dt$$
I trust you can finish from here.
