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**Unformatted text preview: **Lecture 29 18.01 Fall 2006 Lecture 29: Partial Fractions We continue the discussion we started last lecture about integrating rational functions. We defined a rational function as the ratio of two polynomials: P ( x ) Q ( x ) We looked at the example 1 3 x- 1 + x + 2 dx = ln | x- 1 | + 3 ln | x + 2 | + c That same problem can be disguised: 1 + 3 = ( x + 2) + 3( x- 1) = 4 x- 1 x- 1 x + 2 ( x- 1)( x + 2) x 2 + x- 2 which leaves us to integrate this: 4 x- 1 dx = ??? x 2 + x- 2 P ( x ) Goal : we want to figure out a systematic way to split into simpler pieces. Q ( x ) First, we factor the denominator Q ( x ) . 4 x- 1 = 4 x- 1 = A + B x 2 + x- 2 ( x- 1)( x + 2) x- 1 x + 2 Theres a slow way to find A and B . You can clear the denominator by multiplying through by ( x- 1)( x + 2) : (4 x- 1) = A ( x + 2) + B ( x- 1) From this, you find 4 = A + B and- 1 = 2 A- B You can then solve these simultaneous linear equations for A and B . This approach can take a very long time if youre working with 3, 4, or more variables.long time if youre working with 3, 4, or more variables....

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