Overview
This week, we see the last integration techniques we will cover in this course, starting with the method of partial fractions: This technique enables us to antiderive certain ratios of polynomials (i.e. rational functions), for which other methods (like substitution or integration by parts) might not apply. The idea is to replace the rational function by a polymomial plus the sum of simpler rational functions with denominators of at most degree 2.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Be able to do polynomial division in order to write a rational function as a polynomial plus a ratio of polynomials for which the degree of the numerator is less than that of the denominator.
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Decompose a rational function into its partial fractions when the factors of the denominator are linear.
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Use a partial fraction decomposition to integrate a rational function when the factors of the denominator are linear.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Decompose a rational function into its partial fractions when some of the factors of the denominator are irreducible quadratics.
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Use a partial fraction decomposition to integrate a rational function when some of the factors of the denominator are irreducible quadratics.
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Perform rationalizing substitutions when appropriate and then use the method of partial fractions to evaluate the integral.
To prepare for class
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Read subsection 5.5.1 in Active Calculus: The Method of Partial Fractions.
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Watch at least one of the following two videos (by NancyPi and Mathématiques BdeB, respectively) which review long division of polynomials: Since the notation for this algorithm is slightly different in French and English, you may want to rather watch the second video if you went to a French high school. Both notations are equally acceptable, just pick one and stick to it.
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Watch the following video (by GVSUmath) which shows how Partial Fraction Decomposition can be used to integrate a ratio of polynomials:
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Watch the first half (until 12:02) of following video (by patrickJMT) which explains the general structure of a Partial Fraction Decomposition when the factors of the denominator are linear:
After class
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Watch the second half (from 12:06) of the following video (by patrickJMT) which explains the general structure of a Partial Fraction Decomposition when the denominator has irreducible quadratic factors:
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Watch the following video (by Lorenzo Sadun) which shows a full example of using a Partial Fraction Decomposition with irreducible quadratic factors:
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Watch the following video (by patrickJMT) which shows how a rationalizing substitution can lead to a rational expression which can be integrated using the method of partial fractions:
