Answer
$$ - \frac{{{{\cos }^6}5x}}{{30}} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\cos }^5}5x\sin 5x} dx \cr
& {\text{substitute }}u = \cos 5x, \cr
& du = - \sin 5x\left( 5 \right)dx \cr
& - \left( {1/5} \right)du = \sin 5xdx \cr
& \int {{{\cos }^5}5x\sin 5x} dx = \int {{u^5}\left( { - 1/5} \right)} du \cr
& = - \frac{1}{5}\int {{u^5}du} \cr
& {\text{find the antiderivative}} \cr
& = - \frac{1}{5}\left( {\frac{{{u^6}}}{6}} \right) + C \cr
& = - \frac{{{u^6}}}{{30}} + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \cos 5x, \cr
& = - \frac{{{{\cos }^6}5x}}{{30}} + C \cr} $$
