Answer
$$ - \frac{1}{{4{x^2}}} + \frac{8}{3}{x^{3/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left[ {\frac{1}{{2{x^3}}} + 4\sqrt x } \right]} dx \cr
& {\text{radical properties}} \cr
& \int {\left( {\frac{1}{{2{x^3}}} + 4{x^{1/2}}} \right)} dx \cr
& {\text{negative exponent}} \cr
& \int {\left( {\frac{1}{2}{x^{ - 3}} + 4{x^{1/2}}} \right)} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \frac{1}{2}\left( {\frac{{{x^{ - 2}}}}{{ - 2}}} \right) + 4\left( {\frac{{{x^{3/2}}}}{{3/2}}} \right) + C \cr
& = - \frac{1}{{4{x^2}}} + 4\left( {\frac{2}{3}{x^{3/2}}} \right) + C \cr
& = - \frac{1}{{4{x^2}}} + \frac{8}{3}{x^{3/2}} + C \cr} $$
