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MTH 252 Integral Calculus Chapter 8 – Principles of Integral Evaluation Section 8.5 – Integrating Rational Functions by Partial Fractions Copyright © 2006 by Ron Wallace, all rights reserved.

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Review: Addition/Subtraction of Fractions Note: The equation works both ways! Problem: Find two fractions whose sum/difference is equal to a third given fraction. The product of the denominators of the two fractions will be the denominator of the given fraction.

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Example: Find two fractions whose sum or difference is: Denominator = 35 Possibilities for the other two denominators are: 1 & 35 and 5 & 7 Many solutions, including: 3 & -2 and 1 & 4/5

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Rational Functions Any function of the form, where are polynomials.

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Fundamental Theorem of Algebra Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. Each real root r, gives a factor (x-r). Each complex root +i has a companion root -i. These give a factor: (ax 2 +bx+c). Hence, every polynomial can be written as a product of linear & quadratic factors.

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Some “Easy” Integrals of Rational Functions Let u=2x-3 du=2dx Linear Denominator Let u=2x-3 du=2dx

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Partial Fractions where Q(x) is a product of linear factors & deg( P(x) ) < deg( Q(x) ) Solve this system for A & B. A=-1, B=1 This method can be extended to any number of distinct linear factors.

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Partial Fractions where Q(x) is a product of linear factors & deg( P(x) ) < deg( Q(x) ) Solve this system for A & B. A=2, B=-7 This method can be extended to any power of the denominator and can be combined with the previous method. Repeated Linear Factors

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More “Easy” Integrals of Rational Functions Quadratic Denominator Let u=x-2 du=dx Finish using trig substitutions. Just like the one above!

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More “Easy” Integrals of Rational Functions Quadratic Denominator Complete the square & trig substitution. Let u = x 2 - 4x + 7 du = 2x – 4 dx

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Partial Fractions where Q(x) is a product of a linear factor & a quadratic factor & deg( P(x) ) < deg( Q(x) ) Solve this system for A, B, & C. A=1, B=0, C=1 This method can be extended to any number of distinct linear & quadratic factors.

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Partial Fractions where Q(x) is a product of quadratic factors & deg( P(x) ) < deg( Q(x) ) … and proceed as before! All of these methods can be combined and extended to handle any rational function where you can factor the denominator into a product of linear and quadratic factors. Repeated Quadratic Factors

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where deg( P(x) ) ≥ deg( Q(x) ) Simplify using long division of polynomials. Example:

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