1. 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.

2. 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.

3. 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

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

5. 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.

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

7. 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.

8. 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

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

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

11. 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.

12. 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

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