# Math Help - Show that this integral is divergent

1. ## Show that this integral is divergent

The question and my work are both attached.

Any help would be greatly appreciated!

2. You correctly applied the definition of the improper integral by splitting up the integral into two parts:

$\int_{-\infty}^{\infty}x\ dx=\lim_{t \to \infty}\int_{-t}^{0}x\ dx+\lim_{t \to \infty}\int_{0}^{t}x\ dx$

But you need to evaluate the two limits independently:

$\lim_{t \to \infty}\int_{-t}^{0}x\ dx=\lim_{t \to \infty}(\frac{1}{2}0^2-\frac{1}{2}(-t)^2)=\lim_{t \to \infty}-\frac{1}{2}t^2=-\infty$

and

$\lim_{t \to \infty}\int_{0}^{t}x\ dx=\lim_{t \to \infty}(\frac{1}{2}t^2-\frac{1}{2}(0^2))=\lim_{t \to \infty}\frac{1}{2}t^2=\infty$

You combined the two limits after integrating but before taking the limit, and the $t^2$ terms canceled out.

Post again in this thread if you're still having trouble.