Does $\int_\limits{0}^{\infty} \frac{\sin(2x)}{x}dx $ absolutely converge? My teacher gave me this task to prove it, but I have no idea how to begin.
Can I have any clue?

Note that we can write
$$\begin{align}
\int_0^{N\pi/2}\left|\frac{\sin(2x)}{x}\right|\,dx&=\sum_{n=1}^N \int_{(n-1)\pi}^{n\pi}\frac{|\sin(x)|}{x}\,dx\\\\
&\ge \frac1\pi \sum_{n=1}^N\frac1n \int_{(n-1)\pi}^{n\pi}|\sin(x)|\,dx\\\\
&=\frac2\pi \sum_{n=1}^N \frac1n
\end{align}$$
Inasmuch as the harmonic series diverges, the integral fails to be absolutely convergent.
