Let $N$ be a strictly positive integer. How to integrate $\int_0^N \frac{\vert \sin(x) \vert}{x} \ \mathrm{d}x$ ? I tried to subdivise $[0,N]$ in distincts intervals but couldn't achieve a proper resolution of the integral.
I have to show that the limit when N tend to $\infty$ is $\infty$ , so I would also accept an answer that could permit to show it.
You can consider that $\int_0^{\pi} \sin x \,dx = 2$ and use this result to conclude $$\int_0^{\pi} \frac{|\sin x|}{x}\,dx \geq \frac{1}{\pi}\int_0^{\pi} \sin x\,dx = \frac{2}{\pi}.$$ Now show that similarly $$\int_{(k-1)\pi}^{k\pi}\frac{|\sin x|}{x}\,dx \geq \frac{2}{k\pi}, \quad k \in \mathbb N \setminus \{0\},$$ so that $$\int_0^{\infty} \frac{|\sin x|}{x}\,dx = \sum_{k=1}^{\infty} \int_{(k-1)\pi}^{k\pi} \frac{|\sin x|}{x}\,dx \geq \frac{2}{\pi}\sum_{k=1}^{\infty}\frac{1}{k} = \infty.$$
