Objectives: In this tutorial, we consider the problem where we are given the derivative F of some function f and we want to find the function f. We call f the antiderivative or indefinite integral of F. After looking at some examples, we discuss the relationship between two antiderivatives for the same function. We define the notion of initial conditions. Some basic properties of indefinite integrals are stated with examples provided. Applications of indefinite integrals are also provided. After working through these materials, the student should be able

• to derive the integrals in the Table of Elementary Integrals;
• to evaluate simple integrals using the Table of Elementary Integrals;
• to evaluate simple integrals with initial conditions.

Modules:

 Definition. A function F is an antiderivative or an indefinite integral of the function f if the derivative F' = f. We use the notation to indicate that Fis an indefinite integral of f. Using this notation, we have if and only if • Example
[You can get more examples if you continue to click on this link.]

 Uniqueness Theorem. If F and G are antiderivatives of f on some interval I ( i.e., F'(x) = G'(x) = f(x) for all x in I ) then there is a constant C such that F(x) = G(x) + C for all x in I. As a consequence of this theorem, we will usually add the constant C to an indefinite integral: • Discussion [Using Flash]
 Inverse property of indefinite integrals:  