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# lec29 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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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
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lec29 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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