Question:

Let $I = \int \frac{2x - \sqrt{\sin^{-1} x}}{\sqrt{1-x^2}} dx = \int \frac{2x dx}{\sqrt{1-x^2}} - \int \frac{\sqrt{\sin^{-1} x}}{\sqrt{1-x^2}} dx = \int \frac{-dt}{\sqrt{t}} - \int \sqrt{u} du \left[ \begin{matrix} \text{Put } t = 1 - x^2 \\ u = \sin^{-1} x \end{matrix} \right] = -2\sqrt{t} - \frac{2}{3} u^{3/2} + C = -2\sqrt{1-x^2} - \frac{2}{3} (\sin^{-1} x)^{3/2} + C$