Finding Antiderivatives
Finding Antiderivatives
In order to use Part 2 of the Fundamental Theorem of Calculus, an antiderivative of the integrand is needed, but sometimes it is not easy to find one. This section collects some of the information we already have about the general properties of antiderivatives and about antiderivatives of particular functions. It shows how to use the information you already have to find antiderivatives of more complicated functions and introduces a "change of variable" technique to make this job easier.
INDEFINITE INTEGRALS AND ANTIDERIVATIVES
Antiderivatives appear so often that there is a notation to indicate the antiderivative of a function:
Definition: , read as "the indefinite integral of
" or as "the antiderivatives of
,"
represents the collection (or family) of functions whose derivatives are
.
If is an antiderivative of
, then every member of the family
has the form
for
some constant
. We write
, where
represents an arbitrary constant.
There are no small families in the world of antiderivatives: if has one antiderivative
, then
has an
infinite number of antiderivatives and every one of them has the form
. This means that there are
many ways to write a particular indefinite integral and some of them may look very different. You can
check that
, and
all have the same derivative
, so the indefinite integral of
, can be written in
several ways:
.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.7-Finding-Antiderivatives.pdf This work is licensed under a Creative Commons Attribution 3.0 License.