$\int\limits_{a}^{b} \frac{d x}{\sqrt{(x-a)(b-x)}}=\int_{a}^{b} \frac{1}{\sqrt{-x^{2}+(a+b) x-a b}} d x$
$=\int\limits_{a}^{b} \frac{1}{\sqrt{\left(\frac{b-a}{2}\right)^{2}-\left(x-\frac{a+b}{2}\right)^{2}}} d x$
$=\left[\sin ^{-1}\left(\frac{x-\frac{a+b}{2}}{\frac{b-a}{2}}\right)\right]_{a}^{b}$
$=\sin ^{-1} 1-\sin ^{-1}(-1)$
$=\frac{\pi}{2}+\frac{\pi}{2}=\pi$
In above integral if limits are not given, then it will be solved by substituting $x=a \cos ^{2} \theta+b \sin ^{2} \theta$