MTH 252 Integral Calculus Chapter 6 – Integration Section 6.2 – The Indefinite Integral Copyright © 2005 by Ron Wallace, all rights reserved.

Antiderivative F(x) is an antiderivative of f(x) if F’(x) = f(x). Example:

Antiderivative F(x) is an antiderivative of f(x) if F’(x) = f(x). Five antiderivatives of f(x)=2x-5 w/ c = 0, ±2, ±4

Antiderivative If F’(x) = f(x), and c is any constant, then F(x) + c is an antiderivative of f(x). Therefore: Are there any other antiderivatives of f(x)?

Antiderivative Assume that F’(x) = f(x) and G’(x) = f(x). Then d/dx[F(x) - G(x)] = f(x) - f(x) = 0 Therfore F(x) - G(x) = c So, antiderivatives of a function differ by a constant.

The Indefinite Integral The process of finding an antiderivative is called integration. Notation: “The derivative of F(x)+c is f(x).” “The indefinite integral of f(x) is F(x)+c.” Note that these two statements are different notations for the same fact (just opposite processes).

Integration Formulas Just reverse the differentiation formulas …

Properties of Integration These follow directly from the similar differentiation properties. Examples? See page 363.