for it's
i just realized that what i did is wrong.
note that my inequality doesn't hold for that's a another case which need to be treated appart.
we split the interval into two: and
in order to have convergence, we require the convergence for both integral, but to get a divergence, we just need that one of them be divergent, so, if you use my estimation now for it still holds and the integral diverges without analyzing the first piece.