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

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marked as duplicate by Siminore, Shailesh, José Carlos Santos, Robert Z real-analysis Jul 2 '18 at 10:04

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  • $\begingroup$ You should add your thoughts, and maybe your attempts. $\endgroup$ – Siminore Jul 2 '18 at 9:28
  • $\begingroup$ I tried to make a series and show that that series is divergent but im having some trouble figuring out the right series to consider $\endgroup$ – Pedro Miguel Santos Jul 2 '18 at 9:36
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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 $(2k\pi +\pi /4, 2k \pi +3\pi /4) )$.

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  • $\begingroup$ alright i understand what you are doing , just one question, does |f| not being lebesgue integrable implies that f isnt lebesgue integrable? $\endgroup$ – Pedro Miguel Santos Jul 2 '18 at 9:54
  • $\begingroup$ Yes, unlike Riemann integrals, Lebesgue integrals have the property that $f$ is integrable iff $|f|$ is. $\endgroup$ – Kavi Rama Murthy Jul 2 '18 at 10:07
  • $\begingroup$ Alright , Thanks. $\endgroup$ – Pedro Miguel Santos Jul 2 '18 at 10:07

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