How to integrate $\int_a^b t \cdot \sqrt{\frac{t - a}{b - t}} \,dt$? How to integrate $\displaystyle \int_a^b t \cdot \sqrt{\frac{t - a}{b - t}} \,dt$?
I literally have no idea how to integrate this integral. 
I've tried all basic methods, but it seems like a quite hard problem for a beginner.

$ \frac{t-a}{b-t} $ increases from $0$ to $\infty$ on the interval of integration, so let's try $u^2=\frac{t-a}{b-t}$. Then $t = \frac{a+bu^2}{1+u^2} $, so $dt = \frac{2(b-a)u}{(1+u^2)^2}$. Hence the integral becomes
$$ 2(b-a)\int_0^{\infty} \frac{u^2(a+bu^2)}{(1+u^2)^3} \, du, $$
which can be done using, for example, integration by parts.
