partial_frac - (8.4 Partial Fractions This is an algebraic...

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(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
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or 14 = 7 A so A = 2. Therefore, our partial fraction expansion is 5 x - 1 ( x - 3)( x + 4) = 2 x - 3
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