real analysis - Calculate the integral $int_{0}^{infty}|frac{sin{x}}{x} |dx$

I'm trying to solve this problem from Real Analysis of Folland but can't find any solution for it. Can anyone help me ?. Thanks so much.

$$\mbox{Show that}\quad
\int_{0}^{\infty}\left\vert\,{\sin\left(x\right) \over x}\,\right\vert\,{\rm d}x
=\infty
$$

And also, can we calculate the similar integral
$\int_{0}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x$ ?. Please help me clarify this. I really appreciate.

Answer

$$
\int\limits_0^\infty \left|\frac{\sin x}{x} \right| \mathrm{d}x \\
=\sum\limits_{n = 0}^\infty \int\limits_{n\pi}^{(n+1)\pi} \left|\frac{\sin x}{x} \right| \mathrm{d}x \\
\geq \sum\limits_{n = 0}^\infty \int\limits_{n\pi}^{(n+1)\pi} \left|\frac{\sin x}{(n+1)\pi} \right| \mathrm{d}x \\
= \sum\limits_{n = 0}^\infty \frac{1}{(n+1)\pi}\int\limits_{n\pi}^{(n+1)\pi} \left|\sin x \right| \mathrm{d}x \\
= \sum\limits_{n = 0}^\infty \frac{2}{(n+1)\pi}\\

= \frac{2}{\pi}\sum\limits_{n = 0}^\infty \frac{1}{n+1}\\
= \frac{2}{\pi}\left(1+\frac{1}{2}+\frac{1}{3}+ \dots\right) = \infty
$$