Chapter 6: Techniques of Integration
Section 6.1 - Integration by Parts
Section 6.2 - Trigonometric Integrals
Section 6.3 - Trig Substitutions
Section 6.4 - The Algebra of Partial Fractions
Section 6.5 - Integrating the Fractions in a Partial-Fraction Decomposition
Section 6.6 - Rationalizing Substitutions
Section 6.7 - Numeric Methods
There was a time during the historical development of the calculus when the question "Which functions have antiderivatives expressible in terms of the elementary functions?" was pressing. That question has long since been answered, and a section on the techniques of finding antiderivatives is anticlimactic. The answers are known, so what remains is a review of some of the techniques by means of which some antiderivatives can be found. That is the apparent drift of Chapter 6, and if the chapter is regarded in that light only, several very important ideas will be lost.
The technique of "integration by parts" in Section 6.1 should be looked at not so much as a way to find the antiderivative of a product, but rather, as a device for shifting a derivative from one factor to another in an integrand that is a product. Of course, integration by parts can be used to find an antiderivative that could otherwise be obtained from a table of integrals. But its use as a theoretical tool in such fields as integral transforms, and Sturm-Liouville theory in differential equations is the real reason why the topic matters.
A substitution technique from this chapter will rarely, if ever, be used to evaluate an integral outside the confines of a course in integral calculus. Tools such as a table of integrals, or Maple, would ordinarily be the first recourse for such a task. The residual insight that should be the distillation of this chapter is the ability to see substitutions as a way to change an integral from one form to another, from a form in which its antiderivative isn't known to a form in which it is recognized as a known or tabulated result.
The numeric methods for evaluating definite integrals in Section 6.7 are just the basis for modern numeric integrators. The typical modern numeric integrator, including the default one in Maple, is an adaptive process, one that automatically and continually varies the stepsize to guarantee a given level of accuracy. Students who go on to a numerical analysis course will find tools in the Student NumericalAnalysis package of great help in visualizing how an adaptive integrator works.
Finally, the reader should understand that what Maple is programmed "under the hood" to do when finding an antiderivative is generally not what the student is expected to learn in the integral calculus course. For example, there is a class of integrands for which Maple finds antiderivatives by invoking a theory that declares the antiderivative has to be a sum of functions taken from a particular collection. So, Maple writes a linear combination of these functions, differentiates, and solves algebraic equations for the coefficients. This is definitely not what a student is expected to do, and often is the reason why Maple's antiderivatives are in slightly different forms than textbook results.
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