Answer
$ - \frac{{{{\cos }^4}\theta }}{4} + C$
Work Step by Step
$$\eqalign{
& \int {{{\cos }^3}\theta } \sin \theta d\theta \cr
& {\text{Let }}u = \cos \theta ,{\text{ then }}du = - \sin \theta d\theta \cr
& {\text{Apply the substitution}} \cr
& \int {{{\cos }^3}\theta } \sin \theta d\theta = \int {{u^3}} \left( { - du} \right) \cr
& = - \int {{u^3}} du \cr
& {\text{Integrate apply the power rule }}\int {{u^n}du} = \frac{{{u^{n + 1}}}}{{n + 1}}{\text{ + C }} \cr
& = - \frac{{{u^4}}}{4} + C \cr
& {\text{Write in terms of }}\theta ,{\text{ substitute }}u = \cos \theta \cr
& = - \frac{{{{\cos }^4}\theta }}{4} + C \cr} $$
