Answer
$$3{\left( 5 \right)^{2/3}}$$
Work Step by Step
$$\eqalign{
& \int_0^5 {\frac{2}{{{x^{1/3}}}}} dx \cr
& {\text{The integrand }}\frac{2}{{{x^{1/3}}}}{\text{ is not continuous at }}x = 0,{\text{ then by the}} \cr
& {\text{definition of improper integrals}} \cr
& \int_0^5 {\frac{2}{{{x^{1/3}}}}} dx = \mathop {\lim }\limits_{r \to {0^ + }} \int_r^5 {\frac{2}{{{x^{1/3}}}}} dx \cr
& = 2\mathop {\lim }\limits_{r \to {0^ + }} \int_r^5 {{x^{ - 1/3}}} dx \cr
& {\text{Integrating }} \cr
& = 2\mathop {\lim }\limits_{r \to {0^ + }} \left[ {\frac{{{x^{2/3}}}}{{2/3}}} \right]_r^5 \cr
& = 3\mathop {\lim }\limits_{r \to {0^ + }} \left[ {{x^{2/3}}} \right]_r^5 \cr
& = 3\mathop {\lim }\limits_{r \to {0^ + }} \left[ {{{\left( 5 \right)}^{2/3}} - {r^{2/3}}} \right] \cr
& {\text{Evaluate the limit when }}r \to {0^ + } \cr
& = 3\left[ {{{\left( 5 \right)}^{2/3}} - {{\left( {{0^ + }} \right)}^{2/3}}} \right] \cr
& = 3{\left( 5 \right)^{2/3}} \cr} $$
