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.
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
[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
- Discussion [Using Flash]
Inverse property of indefinite integrals: