On the existence of antiderivatives of some real functions

Abstract

An antiderivative of a real function f(x) defined on an interval I ⊂ R is a function F(x) whose derivative is equal to f(x), that is, F 0 (x) = f(x), for all x ∈ I. Antidifferentiation is the process of finding the set of all antiderivatives of a given function. If f and g are defined on the same interval I, then the set of antiderivatives of the sum of f and g equals the sum of the general antiderivatives of f and g. In general, the antiderivatives of the product of two functions f and g do not coincide to the product of the antiderivatives of f and g. Moreover, the fact that f and g have antiderivatives does not imply that the product f · g has antiderivatives. Our aim in this paper is to present some conditions which ensure that the product f · g and the composition f ◦ g of two functions f and g has antiderivatives.