Answer
$$\frac{{2{x^{3/2}}}}{3} + 9{x^{1/3}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {{x^{1/2}} + 3{x^{ - 2/3}}} \right)} dx \cr
& {\text{distribute the integrand by using the sum rule }} \cr
& \int {\left[ {f\left( x \right) \pm g\left( x \right)} \right]} dx = \int {f\left( x \right)} dx \pm \int {g\left( x \right)} dx \cr
& \cr
& = \int {{x^{1/2}}} dx + \int {3{x^{ - 2/3}}} dx \cr
& {\text{use multiple constant rule }}\int {k \cdot f\left( x \right)} dx = k\int {f\left( x \right)} dx \cr
& = \int {{x^{1/2}}} dx + 3\int {{x^{ - 2/3}}} dx \cr
& {\text{use power rule }}\int {{x^x}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr
& = \frac{{{x^{1/2 + 1}}}}{{1/2 + 1}} + 3\left( {\frac{{{x^{ - 2/3 + 1}}}}{{ - 2/3 + 1}}} \right) + C \cr
& {\text{simplifying}} \cr
& = \frac{{{x^{3/2}}}}{{3/2}} + 3\left( {\frac{{{x^{1/3}}}}{{1/3}}} \right) + C \cr
& = \frac{{2{x^{3/2}}}}{3} + 9{x^{1/3}} + C \cr} $$
