Answer
$$ - \frac{{3{{\cos }^{4/3}}x}}{4} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\cos }^{1/3}}x\sin x} dx \cr
& {\text{substitute }}u = \cos x,{\text{ }}du = - \sin xdx \cr
& \int {{{\cos }^{1/3}}x} \sin xdx = \int {{u^{1/3}}\left( { - du} \right)} \cr
& = - \int {{u^{1/3}}du} \cr
& {\text{find the antiderivative by the power rule}} \cr
& = - \frac{{{u^{4/3}}}}{{4/3}} + C \cr
& = - \frac{{3{u^{4/3}}}}{4} + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \cos x \cr
& = - \frac{{3{{\left( {\cos x} \right)}^{4/3}}}}{4} + C \cr
& = - \frac{{3{{\cos }^{4/3}}x}}{4} + C \cr} $$
