Chapter 6
Techniques of
Integration
In this chapter, we expand our repertoire for antiderivatives beyond the “elementary” func-
tions discussed so far. A review of the table of elementary antiderivatives (found in Chap-
ter
??
) will be useful. Here we will discuss a number of methods for finding antiderivatives.
We refer to these collected “tricks” as methods of integration. As will be shown, in some
cases, these methods are systematic (i.e. with clear steps), whereas in other cases, guess-
work and trial and error is an important part of the process.
A primary method of integration to be described is
substitution
. A close relationship
exists between the chain rule of differential calculus and the substitution method. A second
very important method is
integration by parts
. Aside from its usefulness in integration
per se, this method has numerous applications in physics, mathematics, and other sciences.
Many other techniques of integration used to form a core of methods taught in such courses
in integral calculus.
Many of these are quite technical.
Nowadays, with sophisticated
mathematical software packages (including Maple and Mathematica), integration can be
carried our automatically via computation called “symbolic manipulation”, reducing our
need to dwell on these technical methods.
6.1
Differential notation
We begin by familiarizing the reader with notation that appears frequently in substitution
integrals, i.e. differential notation. Consider a straight line
y
=
mx
+
b.
Recall that the slope of the line,
m
, is
m
=
change in
y
change in
x
=
Δ
y
Δ
x
.
This relationship can also be written in the form
Δ
y
=
m
Δ
x.
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Chapter 6. Techniques of Integration