Write a value

let $\sin x=t$

Differentiating on both sides we get,

$\cos x d x=d t$

Substituting above equation in $\int \sin ^{3} x \cos x d x$ we get,

$=\int t^{3} d t$

$=\frac{t^{4}}{4}+c$

$=\frac{\sin ^{4} x}{4}+c$