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Integration or Evaluation Bug


I have found what is either a bug in integration or evaluation of
exponential integrals and boiled it down to this simple example

NIntegrate[Exp[-w] Cos[w t] / (w^2+1),{w,0,1000}]

Plotted as a function of t, this is correctly oscillatory decay

Assuming[t > 1/1000, Integrate[Exp[-w] Cos[w t] / (w^2+1),{w,0,1000}]]

Plotted as a function of t, this is incorrectly growth instead of
decay.

The analytic answer has all terms like Exp[stuff t] Ei[-stuff t]
If stuff was real, that would decay like 1/(stuff t)

So I don't know if it's an integration bug or a bug in Ei evaluation


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