What kind of integration? Riemann, Lebesgue, something else?
For Riemann integration, enumerate the rationals in the interval as , then let
Then for every finite n, but the limit function is not integrable.
I don't think that from the assumption that f be non-negative, non-integrable it follows that is integrable.
For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of ), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.
So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.