<s>[INST] Show that $\int_{-\infty}^{+\infty}\left|\frac{\sin x}{x}\right|\mathrm dx$ diverges to infinity Show that  integration of $$\int_{-\infty}^{+\infty}\left|\dfrac{\sin x}{x}\right|\mathrm dx$$ is equal to infinity.
Is  the limit $$\lim_{x\to\infty}\frac{\sin x}{x} =1$$  useful here?
 [/INST] Let $$\mathcal{A}=\left\{ x\in\mathbb{R}_{>0}:|\sin x| \ge  \frac{1}{2} \right\}  \subset \bigcup_{n\in\mathbb{N}}\left[ \pi n+ \frac{\pi }{6} , \pi n+ \frac{5 \pi }{6}\right]$$ then $$\int_{\mathbb{R}}\left|  \frac{\sin x}{x} \right| \text{d}x   \ge  \int_{\mathbb{R}_{>0}}\left|  \frac{\sin x}{x} \right| \text{d}x  \ge  \int_{\mathcal{A}}\left|  \frac{\sin x}{x} \right| \text{d}x  \ge   \frac{1}{2} \int_{\mathcal{A}}  \frac{1}{x}\text{d}x  \ge   \frac{1}{2} \sum_{n\in\mathbb{N}} \int_{\pi n+ \frac{\pi }{6}}^{\pi n+ \frac{5\pi }{6}}  \frac{1}{x}\text{d}x  $$
but
$$ \frac{1}{2} \sum_{n\in\mathbb{N}} \int_{\pi n+ \frac{\pi }{6}}^{\pi n+ \frac{5\pi }{6}}  \frac{1}{x}\text{d}x=\frac{1}{2}\sum_{n\in\mathbb{N}} \ln \left(  \frac{\pi n+ \frac{5\pi }{6}}{\pi n+ \frac{\pi }{6}} \right)$$
and $$\ln \left(  \frac{\pi n+ \frac{5\pi }{6}}{\pi n+ \frac{\pi }{6}} \right)\sim \frac{\pi}{\pi n+ \frac{\pi }{6}}$$
and $$\sum_{n\in\mathbb{N}} \frac{\pi}{\pi n+ \frac{\pi }{6}}$$
is divergent so the integral too.
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