Answer
$$\frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {3{x^{1/3}} + 4{x^{ - 1/3}} + 6} \right)} dx \cr
& {\text{use power rule for indefinite integrals}} \cr
& = 3\left( {\frac{{{x^{4/3}}}}{{4/3}}} \right) + 4\left( {\frac{{{x^{2/3}}}}{{2/3}}} \right) + 6x + C \cr
& {\text{simplify}} \cr
& = \frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dx}}\left( {\frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C} \right) \cr
& = \frac{9}{4}\left( {\frac{4}{3}} \right){x^{1/3}} + 6\left( {\frac{2}{3}} \right){x^{ - 1/3}} + 6\left( 1 \right) + 0 \cr
& = 3{x^{1/3}} + 4{x^{ - 1/3}} + 6 \cr} $$
