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I just found a curious integral. I wondered if it comes from a bigger group of integral definitions:

[tex]\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)[/tex]

Where Si(x) is the sine integral function [itex]\mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x[/itex]

I proved the equation by developing Si(x) in Taylor series and there is a nice simplification between Si Taylor coefficients and the Gamma function.

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# Curious definite integral : sine integral times exponential

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