Answer
$$3{x^{4/3}} - \frac{{12{x^{7/3}}}}{7} + \frac{{3{x^{10/3}}}}{{10}} + C$$
Work Step by Step
$$\eqalign{
& \int {{x^{1/3}}{{\left( {2 - x} \right)}^2}dx} \cr
& {\text{expand integrand}} \cr
& = \int {{x^{1/3}}\left( {4 - 4x + {x^2}} \right)dx} \cr
& {\text{multiply}} \cr
& = \int {\left( {4{x^{1/3}} - 4{x^{4/3}} + {x^{7/3}}} \right)dx} \cr
& = \int {4{x^{1/3}}dx} - \int {4{x^{4/3}}dx} + \int {{x^{7/3}}} dx \cr
& {\text{power rule}} \cr
& = 4\left( {\frac{{{x^{4/3}}}}{{4/3}}} \right) - 4\left( {\frac{{{x^{7/3}}}}{{7/3}}} \right) + \frac{{{x^{10/3}}}}{{10/3}} + C \cr
& = 3{x^{4/3}} - \frac{{12{x^{7/3}}}}{7} + \frac{{3{x^{10/3}}}}{{10}} + C \cr
& \cr
& {\text{check by differentiation}} \cr
& = \frac{d}{{dx}}\left[ {3{x^{4/3}} - \frac{{12{x^{7/3}}}}{7} + \frac{{3{x^{10/3}}}}{{10}} + C} \right] \cr
& = 3\left( {\frac{4}{3}{x^{1/3}}} \right) - \frac{{12}}{7}\left( {\frac{7}{3}{x^{4/3}}} \right) + \frac{3}{{10}}\left( {\frac{{10}}{3}{x^{7/3}}} \right) + 0 \cr
& = 4{x^{1/3}} - 4{x^{4/3}} + {x^{7/3}} \cr} $$
