Answer
$$3{z^{4/3}} - \frac{3}{2}{z^{2/3}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {4{z^{1/3}} - {z^{ - 1/3}}} \right)} dz \cr
& {\text{use power rule for indefinite integrals}} \cr
& = \frac{{4{z^{4/3}}}}{{4/3}} - \frac{{{z^{2/3}}}}{{2/3}} + C \cr
& {\text{simplify}} \cr
& = 3{z^{4/3}} - \frac{3}{2}{z^{2/3}} + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dz}}\left( {3{z^{4/3}} - \frac{3}{2}{z^{2/3}} + C} \right) \cr
& = 3\left( {\frac{4}{3}} \right){z^{1/3}} - \frac{3}{2}\left( {\frac{2}{3}} \right){z^{ - 1/3}} + 0 \cr
& = 4{z^{1/3}} - {z^{ - 1/3}} \cr} $$
