Answer
Diverges
Work Step by Step
$$\eqalign{
& \int_{ - \infty }^{ - 1} {\frac{1}{{\root 3 \of x }}} dx \cr
& {\text{Using the definition of improper integrals }}\left( {{\text{see page 543}}} \right) \cr
& \underbrace {\int_{ - \infty }^b {f\left( x \right)dx = \mathop {\lim }\limits_{t \to - \infty } } \int_t^b {f\left( x \right)} dx}_ \Downarrow \cr
& \int_{ - \infty }^{ - 1} {\frac{1}{{\root 3 \of x }}} dx = \mathop {\lim }\limits_{t \to - \infty } \int_t^{ - 1} {\frac{1}{{\root 3 \of x }}} dx \cr
& {\text{Using radical properties}} \cr
& = \mathop {\lim }\limits_{t \to - \infty } \int_t^{ - 1} {\frac{1}{{{x^{1/3}}}}} dx \cr
& = \mathop {\lim }\limits_{t \to - \infty } \int_t^{ - 1} {{x^{ - 1/3}}} dx \cr
& {\text{Integrate by using the power rule}} \cr
& = \mathop {\lim }\limits_{t \to - \infty } \left[ {\frac{{{x^{2/3}}}}{{2/3}}} \right]_t^{ - 1} \cr
& = \frac{3}{2}\mathop {\lim }\limits_{t \to - \infty } \left[ {{x^{2/3}}} \right]_t^{ - 1} \cr
& = \frac{3}{2}\mathop {\lim }\limits_{t \to - \infty } \left[ {{{\left( { - 1} \right)}^{2/3}} - {t^{2/3}}} \right] \cr
& {\text{Evaluate the limit when }}t \to - \infty \cr
& = \frac{3}{2}\left[ {{{\left( { - 1} \right)}^{2/3}} - \infty } \right] \cr
& = - \infty \cr
& {\text{Therefore}}{\text{, the integral diverges}} \cr} $$
