Explanation

(c)
Let $I = \int_{0}^{a}\sqrt{\frac{a - x}{x}}dx$
Put $x = {{a\sin}^{2}\theta\ \Rightarrow dx = 2asin\ \theta cos\ \theta d\theta}\$
$\therefore I = \int_{0}^{\pi/2}\sqrt{\frac{\cos^{2}\theta}{\sin^{2}\theta}.2{a\sin}{\theta\cos{\theta d\theta}}}$
$= 2a\int_{0}^{\pi/2}{\cos^{2}{\theta\ d\theta}}$
$= 2a \times \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi a}{2}$