The integral is $\int_1^\infty\frac{\sin{x}}{x}dx$. I know that this integral converges, but I'm wondering if this is valid way to prove it.
This function, if its domain is limited to $\mathbb{N}$, is identical to the sequence: $$-\dfrac{2(-1)^n}{\pi{n}}$$ Because the series $\sum_{k=1}^\infty-\frac{2(-1)^n}{\pi{n}}$ is an alternating series, we know it converges by the alternating series test. By the integral test of convergence, if the series converges, then the integral converges.
What I'm most unsure about is whether or not the integral test works in "reverse". That is, making the integral into a series and using that to determine whether or not the integral converges. Is this a sound proof?
The comparison between the integral and series is valid if the function is monotone or if $$\int_c^\infty |f'(x)|\, dx < \infty$$(see this link). You may be able to get somewhere with the second method. But the best way to go here is Dirichlet's test.