The question and my work are both attached.
Can someone please help me understand what's going on?
Any help would be greatly appreciated!
Thanks in advance!
You correctly applied the definition of the improper integral by splitting up the integral into two parts:
$\displaystyle \int_{-\infty}^{\infty}x\ dx=\lim_{t \to \infty}\int_{-t}^{0}x\ dx+\lim_{t \to \infty}\int_{0}^{t}x\ dx$
But you need to evaluate the two limits independently:
$\displaystyle \lim_{t \to \infty}\int_{-t}^{0}x\ dx=\lim_{t \to \infty}(\frac{1}{2}0^2-\frac{1}{2}(-t)^2)=\lim_{t \to \infty}-\frac{1}{2}t^2=-\infty$
and
$\displaystyle \lim_{t \to \infty}\int_{0}^{t}x\ dx=\lim_{t \to \infty}(\frac{1}{2}t^2-\frac{1}{2}(0^2))=\lim_{t \to \infty}\frac{1}{2}t^2=\infty$
You combined the two limits after integrating but before taking the limit, and the $\displaystyle t^2$ terms canceled out.
Post again in this thread if you're still having trouble.