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Partial Fractions

Learn how to decompose algebraic fractions for easier integration, step-by-step strategies, variations, and examples for tackling complex polynomials. Enhance your integration skills with practical exercises.

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Partial Fractions

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  1. Partial Fractions Lesson 8.5

  2. Partial Fraction Decomposition • Consider adding two algebraic fractions • Partial fraction decomposition reverses the process

  3. Partial Fraction Decomposition • Motivation for this process • The separate terms are easier to integrate

  4. The Process • Given • Where polynomial P(x) has degree < n • P(r) ≠ 0 • Then f(x) can be decomposed with this cascading form

  5. Strategy Given N(x)/D(x) • If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtainDegree of N1(x) will be less than that of D(x) • Now proceed with following steps for N1(x)/D(x)

  6. Strategy • Factor the denominator into factors of the formwhere is irreducible • For each factor the partial fraction must include the following sum of m fractions

  7. Strategy • Quadratic factors: For each factor of the form , the partial fraction decomposition must include the following sum of n fractions.

  8. A Variation • Suppose rational function has distinct linear factors • Then we know

  9. A Variation • Now multiply through by the denominator to clear them from the equation • Let x = 1 and x = -1 • Solve for A and B

  10. What If • Single irreducible quadratic factor • But P(x) degree < 2m • Then cascading form is

  11. Gotta Try It • Given • Then

  12. Gotta Try It • Now equate corresponding coefficients on each side • Solve for A, B, C, and D ?

  13. Even More Exciting • When but • P(x) and D(x) are polynomials with no common factors • D(x) ≠ 0 • Example

  14. Combine the Methods • Consider where • P(x), D(x) have no common factors • D(x) ≠ 0 • Express as cascading functions of

  15. Try It This Time • Given • Now manipulate the expression to determine A, B, and C

  16. Partial Fractions for Integration • Use these principles for the following integrals

  17. Why Are We Doing This? • Remember, the whole idea is tomake the rational function easier to integrate

  18. Assignment • Lesson 8.5 • Page 559 • Exercises 1 – 29 EOO

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