@Solomon Slow But that situation is not Maxwell's demon right? It's the same reasoning as there exists a tiny tiny proabibility that all air molecules in the room place together at the corner and you die because of it, so unlikely but not impossible...

That helped a lot, I thank you :)

That worked! I first tried to compute the integral your way but I got that it diverged to infinity... Later I realized that I had to take Cauchy principal value in order to get the correct result and I got 0, so everything went right in the end :). Thanks a lot!!