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Proof for an integral involving sinc function

How can one calculate $\displaystyle\int_0^{\infty} \frac{\sin^2 x}{x^2} dx$? I know that the answer is $\pi/2$ and I imagine we need to use complex analysis, but I'm not sure how to recast it into a contour integral appropriately.

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Dominic
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    Integrate by parts to see it is equal to $\int_{\mathbb{R}} \frac{\sin x}{x} dx.$ To integrate that use the standard semi-circular contour. – Ragib Zaman Apr 08 '12 at 12:27
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    This can actually also be done without contour integration: $\sin(x)/x$ is (up to a factor) the Fourier transform of $\chi_{[-1,1]}$. You can then use Parseval's identity to evaluate your integral. – Sam Apr 08 '12 at 13:13
  • @Sam L. One sledgehammer changed to another. – Norbert Apr 08 '12 at 13:30
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    @Norbert: Fair enough, but at least two sledge hammers is an improvement on just the one. =) – Sam Apr 08 '12 at 14:03

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