Prove that sin(x)/x is not Lebesgue Integrable in [1,+oo] [duplicate]

Can anyone help me on how to prove that sin(x)/x is not Lebesgue Integrable in [1,+00], Thanks in advance.

marked as duplicate by Siminore, Shailesh, José Carlos Santos, Robert Z real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jul 2 '18 at 10:04

Note that $\int_{\pi /4}^{3 \pi /4} |\sin x| \, dx >0$. Now consider the integrals over $(2k\pi +\pi /4, 2k \pi +3\pi /4) )$ and add up.
Observe that $|x+2k\pi | <2k \pi +3\pi /4$ on $(2k\pi +\pi /4, 2k \pi +3\pi /4) )$.
• Yes, unlike Riemann integrals, Lebesgue integrals have the property that $f$ is integrable iff $|f|$ is. – Kavi Rama Murthy Jul 2 '18 at 10:07