ABSTRACT
M7.1. Indefinite Integral M7.1.1. Antiderivative. Indefinite Integral and Its Properties ◮ Antiderivative. An antiderivative (or primitive function) of a given function f (x) on an interval (a, b) is a differentiable function F (x) such that its derivative is equal to f (x) for all x ∈ (a, b):
since (x2)′ = 2x and (x2 – 1)′ = 2x. THEOREM. Any function f (x) continuous on an interval (a, b) has infinitely many con-
tinuous antiderivatives on (a, b). If F (x) is one of them, then any other antiderivative has the form F (x) + C , where C is a constant. ◮ Indefinite integral. The indefinite integral of a function f (x) is the set, F (x) + C , of all its antiderivatives. This fact is written as∫
f (x) dx = F (x) + C .
