Evaluate the following integrals:

$y=\int \sqrt{\sin x}\left(1-\sin ^{2} x\right) \cos x d x$

Let, $\sin x=t$

Differentiating both side with respect to $x$

$\frac{d t}{d x}=\cos x \Rightarrow d t=\cos x d x$

$y=\int \sqrt{t}\left(1-t^{2}\right) d t$

$y=\int t^{\frac{1}{2}}-t^{\frac{5}{2}} d t$

Using formula $\int t^{n} d t=\frac{t^{n+1}}{n+1}$

$y=\frac{t^{\frac{3}{2}}}{\frac{3}{2}}-\frac{t^{\frac{7}{2}}}{\frac{7}{2}}+c$

Again, put $t=\sin x$

$y=\frac{\sin x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{\sin x^{\frac{7}{2}}}{\frac{7}{2}}+c$