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Calculus II - Notes Integration by Parts

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 section.

 

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.

Calculus II - Notes Integration by Parts

Online Notes / Calculus II (Notes) / Integration Techniques

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