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# ch07 - 140 Chapter 7 Principles of Integral Evaluation...

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Chapter 7: Principles of Integral Evaluation Summary: The primary focus of this chapter is to introduce and explain more advanced integration techniques. With all of the techniques that are introduced, however, the focus is usually to try and find a way to use the basic integral rules (or basic list of antiderivatives) that were introduced in Chapter 5. Many of the ideas basic antiderivatives §5 . 2 in this chapter are methods to use algebraic manipulation or special substitutions so that integrals may be simplified into an integral resembling a basic integral rule. IDEA: At this point the reader should review the integration techniques from Chapter 5. Particular emphasis should be placed upon the basic antideriva- tives ( §5 . 2 ) and u -substitution ( §5 . 3 ). One very powerful method that is introduced in this chapter is the method of in- tegration by parts. This method can simply be thought of as the product rule for antiderivatives. Next various trigonometric integrals are considered and then some special trigonometric substitutions that can be used to simplify a variety of inte- grals that involve square roots and differences and sums of squares. In the case of rational integrands, the method of partial fraction decomposition may sometimes be used to simplify the integrand. In cases where antiderivatives cannot be ex- pressed using simple functions, the idea of numerical integration is considered for definite integrals. Finally at the end of the chapter, integrals are considered that may involve either infinite limits of integration or integrands that have an infinite discontinuity (such as a vertical asymptote). OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. be familiar with the basic integration techniques (see §5 . 2 and §7 . 1) 2. be aware of the different resources for evaluating integrals (§7 . 1) 3. be able to use integration by parts and recognize when it will be useful (§7 . 2) 4. manipulate integrands using trigonometric identities and formulas to obtain more simple integrals (§7 . 3) 5. be able to use trigonometric substitution and recognize when it can be used (§7 . 4) 141
142 6. use partial fraction expansion to evaluate integrals (§7 . 5) 7. use CAS and tables of integrals where appropriate (§7 . 6) 8. use numerical integration techniques where appropriate (§7 . 7 and also §5 . 4) 9. identify definite integrals as improper and determine whether they converge (to a value) or diverge (§7 . 8) 7.1 An Overview of Integration Methods PURPOSE: To review general integration techniques. There are no new ideas in this section but this section does serve as a good check- point for the reader. There are several concepts and methods that the reader should be comfortable with before attempting to learn the material in this chapter. First, REVIEW: 1. basic antiderivatives (§5 . 2) 2. u -substitution (§5 . 2 and §5 . 9) 3. F.T.C (§5 . 6) this section should serve as a reminder to the reader of all of the integration rules that have been introduced in the text so far. It is also a good idea at this point to re-

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ch07 - 140 Chapter 7 Principles of Integral Evaluation...

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