Question

Determine whether integral ${\int }_{-\infty }^{\infty }xdx$ is convergent or divergent.

• A. Convergent at $x>1$
• B. Divergent
• C. Convergent at $x>-1$
• D. Doesn't exist

Answer 1

The correct option is B Divergent

Given : ${\int }_{-\infty }^{\infty }xdx$

${\int }_{-\infty }^{\infty }xdx={\int }_{-\infty }^{0}xdx+{\int }_0^{\infty }xdx=\stackrel{lim}{c\to -\infty }{\int }_c^{0}xdx+\stackrel{lim}{b\to \infty }{\int }_0^{b}xdx=\stackrel{lim}{c\to -\infty }{\left[\frac{x^2}{2}\right]}_c^0+\stackrel{lim}{b\to \infty }{\left[\frac{x^2}{2}\right]}_0^b=\infty$

Hence the integral is divergent.