Answer
$$ - \frac{{{{\cos }^7}3x}}{{21}} + C$$
Work Step by Step
$$\eqalign{
& \int {\sin 3x{{\cos }^6}3x} dx \cr
& u = \cos 3x,{\text{ }}du = - 3\sin 3xdx \cr
& \int {\sin 3x{{\cos }^6}3x} dx = - \frac{1}{3}\int {{u^6}du} \cr
& {\text{find antiderivative}} \cr
& = - \frac{1}{3}\left( {\frac{{{u^7}}}{7}} \right) + C \cr
& = - \frac{{{u^7}}}{{21}} + C \cr
& {\text{replace }}u = \cos 3x \cr
& = - \frac{{{{\left( {\cos 3x} \right)}^7}}}{{21}} + C \cr
& = - \frac{{{{\cos }^7}3x}}{{21}} + C \cr} $$
