Chapter 6.6, Problem 51E

(a) Show that ${\displaystyle {\int }_{-\infty }^{\infty }xdx}$ is divergent.

(b) Show that

$\underset{t\to \infty }{\lim }{\displaystyle {\int }_{-t}^{t}xdx}=0$

This shows that we can’t define

${\displaystyle {\int }_{-\infty }^{\infty }f(x)}dx=\underset{t\to \infty }{\lim }{\displaystyle {\int }_{-t}^{t}f(x)dx}$