# Prove or disprove the convergence of improper integral [duplicate]

Prove or disprove that $$\int_0^\infty \left|\frac{\sin{x}}{x}\right|dx < \infty.$$

I tried by Wolframalpha and this integral seems to satisfy the Cauchy property, but I do not know how to prove it.

Thank you very much.

## marked as duplicate by Siminore, Community♦Feb 13 '17 at 14:38

• Estimate $$\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{x}\,dx.$$ – Daniel Fischer Feb 13 '17 at 14:23
• @DanielFischer Could you explain more specific, please? How could we estimate that integral? – Tien Kha Pham Feb 13 '17 at 14:30
• Find a lower bound so that the series of lower bounds diverges [or an upper bound so that the series of upper bounds converges, but in this case, we have divergence]. – Daniel Fischer Feb 13 '17 at 14:34
• as well as an estimate, you could try and show that the integral below is a lower bound, then show that it is possible to always collect a finite number of these in order to add the same value each time to the total integral - if you are familar with the summation $\Sigma^\infty \frac{1}{n}$ - then consider this integral as lower bound $\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{(k+1)\pi}\,dx$ – Cato Feb 13 '17 at 14:38
• BTW is it ok to say $X < \infty$ – Cato Feb 13 '17 at 14:44