Write a value

let $\cos x=t$

Differentiating on both sides we get,

$-\sin x d x=d t$

Substituting above equation in $\int \cos ^{4} x \sin x d x$ we get,

$=\int-t^{4} d t$

$=-\frac{t^{5}}{5}+c$

$=-\frac{\cos ^{5} x}{5}+c$