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How do you integrate $\int\limits_0^{\infty} e^{i \omega x}\ \mathrm dx$?

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I would use FourierTransform. Since FourierTransform integrates from $-\infty$ to $\infty$, the following will give you the answer (updated per @J.M. comment):

res = FourierTransform[Sqrt[2 Pi] HeavisideTheta[x], x, ω]

I/ω + π DiracDelta[ω]

Let's check if the inverse reproduces the desired theta function:

InverseFourierTransform[res, ω, x]

Sqrt[π/2] (1 + Sign[x])

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    $\begingroup$ FourierTransform[Sqrt[2 π] HeavisideTheta[x], x, ω] ought to work, too. $\endgroup$ – J. M. will be back soon Sep 15 '17 at 22:49
  • $\begingroup$ @J.M. That's what I tried first, but it didn't work for some reason. Maybe lingering definitions. $\endgroup$ – Carl Woll Sep 15 '17 at 22:54
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Mathematica 11.2.0.0 correctly performs

Integrate[Exp[I*ω*x], {x, 0, Infinity}]   

ConditionalExpression[I/ω, Im[ω] > 0]

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