Because the integrand in question is continuous on the interval you're integrating over, it typically isn't necessary to go through the technical details of going back to the full definition of the integral (Riemann, Lebesgue, or otherwise) unless the question specifically asks for it.
In this case, and in the case of most integrals of continuous functions, the easiest way to show existence/nonexistence of integrals is to find an inequality that forces the integral to be finite or that forces the integral to be infinite. I would try to do such an estimate on this integral by noticing that the integrand is finite on the inside of the interval, but blows up near the origin, and then try to use the fact that $x/\sin(x) \rightarrow 1$ as $x \rightarrow 0$. Then you can compare the integral in question to a much simpler integral that you can evaluate more directly.