Partial Fraction Decomposition

1. Partial Fraction Decomposition Integrating rational functions.

2. Integrate the following function…..

3. Now what if we disguise this equation a bit by combining…… If we are given this equation initially: We are going to have to do some expansion in order to put it into a form that is easy to integrate. This process is called PARTIAL FRACTION DECOMPOSITION.

6. The method of Partial Fraction Decomposition ALWAYS works when you are integrating a rational function. Rational Function = Ratio of polynomials You will decompose/expand the rational function so it can be easily integrated.

7. Let Let

8. Let Let *Integrate using U-Substitution

9. Let Let

10. Let • Let

11. rules • Remember is just • If , then • When thinking about your natural log rules, use and to guide you.

12. Which of the following are true? or or

13. Which of the following are true? or or

14. Homework Section 8.5 P. 559 (7-11) Section 5.6 P. 378 (45, 59) Section 5.7 P. 385 (3, 4, 7) Partial Fraction Decomposition • Derivating Arctangent • Integrating Arctangent

16. Let Let Let

18. In all of our examples thus far, the degree of the numerator has been less than the degree of the denominator. If it is the case that the degree of the numerator is greater than or equal to the degree of the denominator, you must reduce using “polynomial” long division. The next few slides will help you to review this technique…….

19. Long “Polynomial” Division Review

23. Since no 𝒙 we must Put 𝟎𝒙.

25. Since no we must Put

26. End of Lesson Homework: Partial Fraction Decomp. Worksheet (14, 15) Orange Book Section 6.5 P. 369 (5, 7, 8, 15-21 odd)

27. Steps to Integrating by PFD • If degree of numerator is greater than or equal to degree of denominator, then use long division to reduce. • Write out or setup the equation as a sum of fractions with unknown numerators. • Solve for the unknown numerators. • Integrate the resulting equation.