Is there a way to compute $$\int_0^\infty \frac{\sin(x)}{x}dx$$ without residue theorem ?
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Related – Error 404 Jul 03 '16 at 09:38
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you can also use Fourier transform on $f(x)=\chi_{[-1/2,1/2]}$. – Surb Jul 03 '16 at 09:52
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use parity and then differentiate $\int_{\mathbb{R}}\frac{\sin(ax)}{x}$. Afterwards make use of the fact that the FT of $1$ is a $\delta$-distribution – tired Jul 03 '16 at 10:11