Answer
$$\frac{6}{7}{x^{7/3}} + 2{x^{1/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {2{x^{4/3}} + {x^{ - 1/2}}} \right)} dx \cr
& {\text{distribute the integrand by using the sum rule }} \cr
& \int {\left[ {f\left( x \right) \pm g\left( x \right)} \right]} dx = \int {f\left( x \right)} dx \pm \int {g\left( x \right)} dx \cr
& \cr
& = \int {2{x^{4/3}}} dx + \int {{x^{ - 1/2}}} dx \cr
& {\text{use multiple constant rule }}\int {k \cdot f\left( x \right)} dx = k\int {f\left( x \right)} dx \cr
& = 2\int {{x^{4/3}}} dx + \int {{x^{ - 1/2}}} dx \cr
& {\text{use power rule }}\int {{x^x}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr
& = 2\left( {\frac{{{x^{4/3 + 1}}}}{{4/3 + 1}}} \right) + \left( {\frac{{{x^{ - 1/2 + 1}}}}{{ - 1/2 + 1}}} \right) + C \cr
& {\text{simplifying}} \cr
& = 2\left( {\frac{{{x^{7/3}}}}{{7/3}}} \right) + \left( {\frac{{{x^{1/2}}}}{{1/2}}} \right) + C \cr
& = \frac{6}{7}{x^{7/3}} + 2{x^{1/2}} + C \cr} $$
