Answer
$$ - \frac{{{{\cos }^4}x}}{4} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\cos }^3}x} \sin xdx \cr
& {\text{substitute }}u = \cos x,{\text{ }}du = - \sin xdx \cr
& \int {{{\cos }^3}x} \sin xdx = \int {{u^3}\left( { - du} \right)} \cr
& = - \int {{u^3}du} \cr
& {\text{find the antiderivative by the power rule}} \cr
& = - \frac{{{u^4}}}{4} + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \cos x \cr
& = - \frac{{{{\left( {\cos x} \right)}^4}}}{4} + C \cr
& = - \frac{{{{\cos }^4}x}}{4} + C \cr} $$
