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lec29 - Lecture 29 18.01 Fall 2006 Lecture 29 Partial...

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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 There’s 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 you’re working with 3, 4, or more variables. There’s a faster way, which we call the “cover-up method”. Multiply both sides by ( x - 1) : 4 x - 1 B x + 2 = A + x + 2 ( x - 1) Set x = 1 to make the B
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