Answer
$${\text{diverges}}$$
Work Step by Step
$$\eqalign{
& \int_1^{ + \infty } {xdx} \cr
& {\text{Using the definition of improper integrals see page 1053}} \cr
& \underbrace {\int_a^{ + \infty } {f\left( x \right)dx = \mathop {\lim }\limits_{M \to + \infty } } \int_a^M {f\left( x \right)} dx}_ \Downarrow \cr
& \int_1^{ + \infty } x dx = \mathop {\lim }\limits_{M \to + \infty } \int_1^M x dx \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{M \to + \infty } \left[ {\frac{{{x^2}}}{2}} \right]_1^M \cr
& = \mathop {\lim }\limits_{M \to + \infty } \left[ {\frac{{{M^2}}}{2} - \frac{{{1^2}}}{2}} \right] \cr
& = \frac{1}{2}\mathop {\lim }\limits_{M \to + \infty } \left[ {{M^2} - 1} \right] \cr
& {\text{Evaluate the limit when }}M \to + \infty \cr
& = \frac{1}{2}\left( {{{\left( \infty \right)}^2} - 1} \right) \cr
& = \infty \cr
& {\text{diverges}} \cr} $$
