Answer
$${\text{diverges}}$$
Work Step by Step
$$\eqalign{
& \int_{ - \infty }^\infty x dx \cr
& {\text{Use the definition }}\int_{ - \infty }^\infty {f\left( x \right)} dx = \int_{ - \infty }^c {f\left( x \right)} dx + \int_c^{ + \infty } {f\left( x \right)} dx \cr
& {\text{Let }}c = 0. \cr
& \int_{ - \infty }^\infty x dx = \int_{ - \infty }^0 x dx + \int_0^{ + \infty } x dx \cr
& {\text{Evaluating the integrals on the right side separately}} \cr
& \int_{ - \infty }^0 x dx = \mathop {\lim }\limits_{a \to - \infty } \int_a^0 x dx \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{{{x^2}}}{2}} \right]_a^0 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{a \to - \infty } \left[ {\frac{{{{\left( 0 \right)}^2}}}{2} - \frac{{{{\left( a \right)}^2}}}{2}} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{{{a^2}}}{2}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Find the limit when }}a \to - \infty \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\left( { - \infty } \right)}^2}}}{2} = + \infty \cr
& {\text{The integral diverges, }}{\text{so the given integral }}\int_{ - \infty }^\infty x dx{\text{ diverges}} \cr} $$
