We consider the following u {\displaystyle u} -substitution: u = sin x , d u = cos x d x {\displaystyle u=\sin x,\;{\rm {d}}u=\cos x{\rm {dx}}} . The integral then becomes
∫ cos x sin x d x = ∫ u 1 / 2 d u = 2 3 u 3 / 2 + C . {\displaystyle \int \cos x{\sqrt {\sin x}}\mathrm {d} x=\int u^{1/2}\mathrm {d} u={\frac {2}{3}}u^{3/2}+C.}
Substituting u = sin x {\displaystyle u=\sin x} back we get:
∫ cos x sin x d x = 2 3 sin 3 / 2 ( x ) + C . {\displaystyle \int \cos x{\sqrt {\sin x}}\mathrm {d} x={\frac {2}{3}}\sin ^{3/2}(x)+C.}
-substitution: 
. The integral then becomes
back we get:
