partial_frac

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

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

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