Improper integral convergence example with absolute value
Solution 1:
Note that for any integer $N>1$
$$\begin{align} \int_1^{N\pi}\left|\frac{\sin(x)}{x}\right|\,dx&\ge \int_\pi^{N\pi}\left|\frac{\sin(x)}{x}\right|\,dx\\\\ &=\sum_{k=1}^{N-1} \int_{k\pi}^{(k+1)\pi}\left|\frac{\sin(x)}{x}\right|\,dx\\\\ &\ge \sum_{k=1}^{N-1} \frac{1}{(k+1)\pi}\int_{k\pi}^{(k+1)\pi}|\sin(x)|\,dx\\\\ &=\frac2\pi\sum_{k=2}^{N}\frac1k \end{align}$$
Inasmuch as the harmonic series diverges, we see that the integral of interest does likewise.