The reason that I claim it is not Riemann integrable is because for any partition P, the upper Riemann sum associated with P is infinity, but its lower Riemann sum is at most 2, so the associated inf & sup taken over all partitions will not agree.
Well, I checked my analysis book, and you're right, Jose. The Lebesgue criteria includes the word "bounded". Moreover, on page 154 of Kirkwood, he defines improper Riemann integrals, where either the interval or the function is unbounded. So all I said about all this is probably twaddle.