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Unformatted text preview: 0.1 Partial Fractions: Distinct Real roots (Simplified case) 0.2 Example: Evaluate Z 2 x + 1 x 2 + 4 x + 3 dx The partial fractions expansion is 2 x + 1 x 2 + 4 x + 3 = a x + 1 + b x + 3 Multiplying through by x 2 + 4 x + 3 = ( x + 1)( x + 3) gives 2 x + 1 = a ( x + 3) + b ( x + 1) Setting x = 1 gives 2 a = 1. Setting x = 3 gives 2 b = 5. Thus we have 2 x + 1 x 2 + 4 x + 3 = 1 2( x + 3) + 5 2( x + 1) and Z 2 x + 1 x 2 + 4 x + 3 dx = Z 1 2( x + 3) + 5 2( x + 1) = 1 2 ln  x + 3  + 5 2 ln  x + 1  + C 0.3 Partial Fractions: Complex Roots The final case to consider is the case when the polynomial has complex roots. We are assuming that the polynomial has real coefficients. In this case the complex roots appear in complex conjugate pairs. Lets assume that there are r real roots and n r complex roots (note that n r must be even). In this case the polynomial admits a factorization of the form Q ( x ) = A n ( x x 1 )( x x 2 ) . . . ( x x r )( x 2 + B 1 x + C 2 )( x 2 + B 2 x + C...
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This note was uploaded on 04/07/2008 for the course MATH 231 taught by Professor Bronski during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 Bronski
 Math, Calculus, Fractions

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