Integral converges/diverges Question:
$$p \in \Bbb R  \int _0 ^\infty \frac {\sin x}{x^p}dx$$
For what values of p does this integral converge/diverge?
Thoughts
We've tried using Dirichlet criteria and bounding it from below, after splitting it from $[0,1]$ and $[1.\infty]$ There seems to be some kind of trick we are probably missing.

For $p = 1$, it is convergent. You can show it by using.
$$
\int_{1}^{b}\frac{\text{sin}\;x}{x}dx = -\frac{\text{cos}\; b}{b} + \text{cos}\;1 - \int_{1}^{b}\frac{\text{cos}\;x}{x^2}dx
$$
The point is, you can show by using integral by parts. 
