I'm trying to work through a couple improper integral problems to study for a final, but am somewhat stuck on showing that $\displaystyle \int^1_0 \frac{\sin x -x}{x^3}$ converges.
Edit: Can you use the weighted mean value theorem for integrals? If I can show that $\sin x -x \leq 1$ on $[0,1]$, then it seems like by that theorem, $\displaystyle \int^1_0 \frac{\sin x -x}{x^3} = \frac{1}{c^3} \int^1_0 \sin x -x$ for some $c \in [0,1]$. This should converge, if I can guarantee that $c \neq 0$.