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Unformatted text preview: (8.4) Partial Fractions This is an algebraic technique for splitting polynomial fractions apart into sums of simple fractions. This technique is used in various places in mathematics; here, we use it for integration. Example. We wish to find 4 x 2 8 x + 15 dx . The idea is simple. It happens that 4 x 2 8 x + 15 = 2 x 5 2 x 3 (notice that x 2 8 x + 15 = ( x 5)( x 3)). Therefore, 4 x 2 8 x + 15 dx = 2 x 5 2 x 3 dx = 2 ln  x 5   2 ln  x 3  + C Of course, we need to know how the polynomial fraction was broken apart. This will be illustrated in the next example. Example. 5 x 1 x 2 + x 12 dx . Here, we want to find numbers A and B such that 5 x 1 x 2 + x 12 = 5 x 1 ( x 3)( x + 4) = A x 3 + B x + 4 . The trick is simple: clear fractions by multiplying the second equation by ( x 3)( x + 4). We obtain: ( x 3)( x + 4) 5 x 1 ( x 3)( x + 4) = ( x 3)( x + 4) A x 3 + B x + 4 = A ( x 3)( x + 4) x 3 + B ( x 3)( x + 4) x + 4 so we get 5 x 1 = A ( x + 4) + B ( x 3) . This enables us to grab ahold of A and B . Let x = 4. The equation becomes 5( 4) 1 = A ( 4 + 4) + B ( 4 3) so that 21 = 7 A and A = 3. Now let x = 3. Then the equation becomes 5(3) 1 = A (3 + 4) + B (3 3) 1 or 14 = 7 A so A = 2. Therefore, our partial fraction expansion is= 2....
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 Fall '08
 VALDIMARSSON
 Algebra, Fractions

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