Noun
improper integral (plural improper integrals)
(mathematics) An integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity.
An improper integral occurs when one or more of these conditions is not satisfied. Source: Internet
Since the integrand is a non-negative real-valued function, there is no difference between having a finite Lebesgue integral and having a finite improper integral (as there is say for the function sin(x)/x when integrated over the entire real line). Source: Internet
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity. Source: Internet