int xcdot sqrt [3](x^2)dx (3x^2cdot sqrt [3](x^2))/(8)+C 0.375sqrt [3](x^8)+C (3x^frac (8)/(3))(8)+C

To solve the integral $\int x\cdot \sqrt [3]{x^{2}}dx$, we can rewrite the expression as $\int x^{\frac{5}{3}}dx$.<br /><br />Using the power rule for integration, we have:<br /><br />$\int x^{\frac{5}{3}}dx = \frac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1} + C$<br /><br />Simplifying the exponent, we get:<br /><br />$\frac{x^{\frac{8}{3}}}{\frac{8}{3}} + C$<br /><br />Multiplying the numerator and denominator by 3, we obtain:<br /><br />$\frac{3x^{\frac{8}{3}}}{8} + C$<br /><br />Therefore, the correct answer is $\frac{3x^{\frac{8}{3}}}{8} + C$.