Solve the integral $I$ = $\int_0^{\infty} \frac{\sin x}{x} dx$
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See my answer here: https://math.stackexchange.com/questions/2506224/confirm-that-int-0-inftyt-1-sin-t-dt-pi-2/2506265#2506265 – aleden Nov 22 '17 at 11:43
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https://math.stackexchange.com/questions/1277916/proving-int-mathbb-r-frac-sinxxdx-pi?noredirect=1&lq=1 – Guy Fsone Nov 22 '17 at 12:20
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G.H. Hardy wrote a paper with several methods of evaluating this integral. See http://math.harvard.edu/~ctm/home/text/others/hardy/sinx/sinx.pdf – awkward Nov 22 '17 at 13:00