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7.4 Integration of Rational Functions By Partial Fractions

7.4 Integration of Rational Functions By Partial Fractions. Example 1. This would be a lot easier if we could re-write it as two separate terms. Multiply by the common denominator. Set like-terms equal to each other. Solve two equations with two unknowns. This technique is called

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7.4 Integration of Rational Functions By Partial Fractions

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  1. 7.4 Integration of Rational Functions By Partial Fractions

  2. Example 1 This would be a lot easier if we could re-write it as two separate terms. Multiply by the common denominator. Set like-terms equal to each other. Solve two equations with two unknowns.

  3. This technique is called Partial Fractions Solve two equations with two unknowns.

  4. The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside. Multiply by the common denominator. Let x= - 1 Let x= 3

  5. Method of Partial Fractions • If the fraction is improper, use long division to rewrite it as a sum of a polynomial and a proper fraction. • If the fraction is proper, factor the denominator completelyand write it as a sum of fractions as follows: a) For each linear factors (ax+b)n, the decomposition must have the form: b) ) For each irreducible quadratic factors (ax2+bx+c)n, the decomposition must have the form:

  6. Example 2 Repeated roots: we must use two terms for partial fractions.

  7. Example 3 If the degree of the numerator is higher than the degree of the denominator, use long division first. (from example one)

  8. Example 4 irreducible quadratic factor repeated root

  9. Examples

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