How to prove that $\frac{\sin x}{x}$ is not Lebesgue integrable on $[0,+\infty]$?

Question

How to prove that $\displaystyle\int_0^{+\infty}\left|\dfrac{\sin x}{x}\right| \, dx = +\infty$ ? Could any one give some hint ? Thanks.

Answer 1

$$\int_{k\pi}^{(k+1)\pi}\left|\frac{\sin x}{x}\right|dx\geq\frac{1}{(k+1)\pi}\int_{k\pi}^{(k+1)\pi}|\sin x|dx.$$

You can bound the integral below by a constant multiple of the harmonic series.

Answer 2

Hint: $\sum \frac {1}{n} =$ is unbounded. Consider the intervals where $|\sin x| > \frac {1}{2}$.