@JonasTeuwen When i solved the improper integrals like you did, I never explicity mentioned the second term like you have mentioned in your answer. Do you think it is necessary to do so? Do you think
\begin{equation}
\lim_{y \to \infty} \left (\int_{[y/\pi]\pi}^y f(x) \, \text{d}x \right ).
\end{equation}

may not converge to zero?

Hmm, *methinks* the thing is like
if the improper integral exits, then the term goes to zero,
else the term may not go to zero, but then the actual improper integral itself does not converge.

$$
\int_0^N\frac{\sin(x)}{x}\,\mathrm{d}x = \left.\frac{1-\cos(x)}{x}\right]_0^N+\int_0^N\frac{1-\cos(x)}{x^2}\,\mathrm{d}x
$$