$$ \int \frac {\sqrt{x}} {\sqrt{1-x}} dx $$ $$ = \int \frac {2 u^2} { \sqrt { 1- u^2}} du \quad (x = u^2) $$ $$ = \int \frac { 2 \sin^2 \theta} { \sqrt { 1- \sin ^2\theta}} \cos \theta d \theta \quad (u = \sin \theta) $$ $$ = \int 2 \sin^2 \theta d \theta = \int (1-\cos (2 \theta)) d \theta $$ $$ = \theta - (1/2) \sin (2 \theta) + c = \theta - \sin \theta \cos \theta +c $$ $$ = \arcsin u - u \sqrt{1 - u^2} +c $$ $$ = \arcsin (\sqrt{x}) - \sqrt{x}\sqrt{1-x} + c $$