Answer
$\frac{{3{x^{4/3}}}}{4} + \frac{{3{x^{2/3}}}}{2} + C$
Work Step by Step
$$\eqalign{
& \int {\left( {\root 3 \of x + \frac{1}{{\root 3 \of x }}} \right)} dx \cr
& {\text{Rewrite the integrand using the radical properties}} \cr
& = \int {\left( {{x^{1/3}} + \frac{1}{{{x^{1/3}}}}} \right)} dx \cr
& = \int {\left( {{x^{1/3}} + {x^{ - 1/3}}} \right)} dx \cr
& {\text{Use the sum rule for integration}} \cr
& = \int {{x^{1/3}}} dx + \int {{x^{ - 1/3}}} dx \cr
& {\text{Use the power rule for integration }}\int {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr
& = \frac{{{x^{1/3 + 1}}}}{{1/3 + 1}} + \frac{{{x^{ - 1/3 + 1}}}}{{ - 1/3 + 1}} + C \cr
& {\text{Simplifying}} \cr
& = \frac{{{x^{4/3}}}}{{4/3}} + \frac{{{x^{2/3}}}}{{2/3}} + C \cr
& = \frac{{3{x^{4/3}}}}{4} + \frac{{3{x^{2/3}}}}{2} + C \cr} $$
