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Problem with NIntegrate -- Monte Carlo


I have a rather complicated function that Mathematica can't seem to handle
with
NIntegrate, Method->MonteCarlo.

This is not my function, but it shows the essence of the problem.
Suppose we have

   ComplicatedF[x_] :=  Re [NIntegrate [E^(- (I x+y+I)^2), {y,0,1}] ]

Then, if you try:

NIntegrate[ComplicatedF[x], {x,0,2}, Method->MonteCarlo]

Mathematica will complain that the integrand is not numerical.
But of course it is numerical and removing the Method->MonteCarlo option
will generate an answer.

Now my actual function is quite complicated which is why
I want to try the MonteCarlo method. Any suggestions?



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