Evaluate $\int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx$

Question

Evaluate: $$ \space \space \int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx $$

Please give proper directions/hints to evaluate this.

Answer 1

You can simply let $t = \arcsin(\sqrt{x})$. Then $dt = \frac{1}{2\sqrt{1-x} \cdot \sqrt{x}} dx$. So we end up wanting to deal with

$$\int 2t \sin t \ dt$$

Now do integration by parts with $u = t, dv = \sin t \ dt$.

Answer 2

We pose $x=\cos^2\theta$ and we have $$ I= \int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}dx=-2\int\theta\cos\theta d\theta, $$ and by integration by parts we have $$I=-2\theta\sin\theta-2\cos\theta+C,$$ Finally we return to the variable $x$ to find $$I=-2\sqrt{x}-2\arcsin(\sqrt{x})\sqrt{1-x}+C.$$