Proper and Improper Riemann Integral in a Single Denition

Abstract

This work proposes a new form of integral which arises from innite partitions. It uses upper and lower series instead of upper and lower Darboux nite sums. It is shown that every Riemann integrable function, both proper and improper, is integrable in the sense proposed here and both integrals have the same value. Furthermore it is shown that the Riemann integral and our integral are equivalent for bounded functions in bounded intervals. The advantage of this new integral is that a single denition allows the integration of bounded or unbounded functions, in bounded or unbounded intervals. The present integral is different from the ordinary Riemann integral, where it is necessary to have the prior denition of bounded functions in bounded intervals.