Question:
Evaluate $\int \sqrt{\sin x} \cos ^{3} x d x$
Solution:
$y=\int \sqrt{\sin x}\left(1-\sin ^{2} x\right) \cos x d x$
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 $t=\sin x$
$y=\frac{\sin x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{\sin x^{\frac{7}{2}}}{\frac{7}{2}}+c$