Answer
$$6$$
Work Step by Step
$$\eqalign{
& \int_0^8 {\frac{{dx}}{{\root 3 \of x }}} \cr
& {\text{The integrand is not defined for }}x = 0,{\text{ then}} \cr
& \int_0^8 {\frac{{dx}}{{\root 3 \of x }}} = \mathop {\lim }\limits_{a \to {0^ + }} \int_a^8 {\frac{{dx}}{{\root 3 \of x }}} \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{a \to {0^ + }} \left[ {\frac{{{x^{2/3}}}}{{2/3}}} \right]_a^8 \cr
& = \frac{3}{2}\mathop {\lim }\limits_{a \to {0^ + }} \left[ {{x^{2/3}}} \right]_a^8 \cr
& = \frac{3}{2}\mathop {\lim }\limits_{a \to {0^ + }} \left[ {{8^{2/3}} - {a^{2/3}}} \right] \cr
& {\text{Evaluating the limit}} \cr
& = \frac{3}{2}\left[ {4 - {{\left( 0 \right)}^{2/3}}} \right] \cr
& = 6 \cr} $$
