Question

Evaluate $\int \sqrt{\frac{x}{a-x}}dx$

Answer 1

Put $x=a{\sin }^2\theta ⟹dx=2a\sin \theta \cos \theta d\theta$

$\int \sqrt{\frac{x}{a-x}}dx=\int \sqrt{\frac{a{\sin }^2\theta }{a-a{\sin }^2\theta }}\left(2a\sin \theta \cos \theta \right)d\theta =2a\int {\sin }^2\theta d\theta =a\int (1-\cos 2\theta )d\theta =a\theta -\frac{a}{2}\sin 2\theta +c$

$=a{\text{sin}}^{-1}(\sqrt{\frac{x}{a}})-a\sqrt{x/a}\times \sqrt{1-x/a}+c=a{\text{sin}}^{-1}(\sqrt{\frac{x}{a}})-\sqrt{ax-x^2}+c$