Answer
$$6$$
Work Step by Step
$$\eqalign{
& \int_0^8 {\frac{{dx}}{{\root 3 \of x }}} \cr
& {\text{The function is undefined for }}x = 0,{\text{ so the integral can be represented as}} \cr
& \int_0^8 {\frac{{dx}}{{\root 3 \of x }}} = \mathop {\lim }\limits_{k \to {0^ + }} \int_k^8 {\frac{{dx}}{{\root 3 \of x }}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to {0^ + }} \int_k^8 {{x^{ - 1/3}}} dx \cr
& {\text{Integrate}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to {0^ + }} \left[ {\frac{{{x^{2/3}}}}{{2/3}}} \right]_k^8 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\mathop {\lim }\limits_{k \to {0^ + }} \left[ {{x^{2/3}}} \right]_k^8 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\mathop {\lim }\limits_{k \to {0^ + }} \left[ {{{\left( 8 \right)}^{2/3}} - {k^{2/3}}} \right]_k^8 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\mathop {\lim }\limits_{k \to {0^ + }} \left[ {4 - {k^{2/3}}} \right] \cr
& \,{\text{Calculate the limit when }}k \to {0^ + } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\left[ {4 - {{\left( {{0^ + }} \right)}^{2/3}}} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\left( 4 \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6 \cr
& {\text{Then}}{\text{,}} \cr
& \int_0^8 {\frac{{dx}}{{\root 3 \of x }}} = 6 \cr} $$
