In this chapter we are going to be looking at various
integration techniques. There are a fair
number of them and some will be easier than others. The point of the chapter is to teach you
these new techniques and so this chapter assumes that you’ve got a fairly good
working knowledge of basic integration as well as substitutions with
integrals. In fact, most integrals
involving “simple” substitutions will not have any of the substitution work
shown. It is going to be assumed that
you can verify the substitution portion of the integration yourself.
Also, most of the integrals done in this chapter will be
indefinite integrals. It is also assumed
that once you can do the indefinite integrals you can also do the definite
integrals and so to conserve space we concentrate mostly on indefinite
integrals. There is one exception to
this and that is the Trig Substitution section and in this case there are some
subtleties involved with definite integrals that we’re going to have to watch
out for. Outside of that however, most
sections will have at most one definite integral example and some sections will
not have any definite integral examples.
Here is a list of topics that are covered in this chapter.
Integration by Parts Of all the integration techniques covered in
this chapter this is probably the one that students are most likely to run into
down the road in other classes.
Integrals Involving Trig Functions In this section we look at integrating certain
products and quotients of trig functions.
Trig Substitutions Here we will look using substitutions
involving trig functions and how they can be used to simplify certain integrals.
Partial Fractions We will use partial fractions to allow us to
do integrals involving some rational functions.
Integrals Involving Roots We will take a look at a substitution that
can, on occasion, be used with integrals involving roots.
Integrals Involving Quadratics In this section we are going to look at some
integrals that involve quadratics.
Integration Strategy We give a general set of guidelines for
determining how to evaluate an integral.
Improper Integrals We will look at integrals with infinite
intervals of integration and integrals with discontinuous integrands in this
Comparison Test for Improper Integrals Here we will use the Comparison Test to
determine if improper integrals converge or diverge.
Approximating Definite Integrals There are many ways to approximate the value
of a definite integral. We will look at
three of them in this section.