\[
\int\limits_0^1 {\frac{{3x^2  + 2}}{{x^{2/3} }}dx}  = \int\limits_0^1 {3x^{2 - \frac{2}{3}} dx}  + \int\limits_0^1 {2x^{ - \frac{2}{3}} dx}  = 3\int\limits_0^1 {x^{\frac{4}{3}} dx}  + 2\int\limits_0^1 {x^{ - \frac{2}{3}} dx}  = \left. {3\frac{{x^{\frac{4}{3} + 1} }}{{\frac{4}{3} + 1}}} \right|_0^1  + \left. {2\frac{{x^{ - \frac{2}{3} + 1} }}{{ - \frac{2}{3} + 1}}} \right|_0^1  = \left. {\frac{9}{7}x^{\frac{7}{3}} } \right|_0^1  + \left. {6x^{\frac{1}{3}} } \right|_0^1  = \frac{9}{7}\left( {1^{\frac{7}{3}}  - 0^{\frac{7}{3}} } \right) + 6\left( {1^{\frac{1}{3}}  - 0^{\frac{1}{3}} } \right) = \frac{9}{7}\left( {1 - 0} \right) + 6\left( {1 - 0} \right) = \frac{9}{7} + 6 = \frac{{51}}{7}
\]