A theory of divergent integrals

We construct a theory of divergent integrals, based on the following definition: we say that ff is (c,α)(c,α) integrable, α>−1α>−1, if, for some a∈Ra∈R, limT→∞∫0T(1−tT)αf(t)dt=a. As a consequence of this definition, some theorems are proved. We study, in detail, the case α=1α=1.