A theory of divergent integrals

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