2-7 - Q is a product of distinct linear factors i.e Q x = a...

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February 7, 2005 Announcements Reading for the week: § 7.4, § 7.5 and § 7.7. Homework #5 (Week 5 Problems) (Covers § 7.1, 7.2 and 7.3; see web for assignment) Collected Tomorrow: Tuesday, February 8. Today § 7.4 Integration of Rational Functions by Partial Fractions. 1
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A rational function is a function such that f ( x ) = P ( x ) Q ( x ) where P ( x ) and Q ( x ) are polynomials, i.e. P ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 0 Q ( x ) = b m x m + b m - 1 x m - 1 + · · · + b 0 with a n = 0 and b m = 0. The degree of P is n and the degree of Q is m . Notation: deg ( P ) = n and deg Q = m . If deg ( P ) deg Q divide P by Q to obtain a remainder R such that deg ( R ) < deg Q , i.e. f ( x ) = P ( x ) Q ( x ) = S ( x ) + R ( x ) Q ( x ) , where R and S are also polynomials.
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Steps to integrate a rational function of the form R ( x ) Q ( x ) with deg ( R ) < deg Q. Step 1 : Factor Q as a product of factors of the form ax + b or ax 2 + bx + c with b 2 - 4 ac < 0 . Step 2 : Express R ( x ) Q ( x ) as a sum of partial frac- tions of the form A ( ax + b ) j or Ax + B ( ax 2 + bx + c ) i , with b 2 - 4 ac < 0.
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Case I:
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Unformatted text preview: Q is a product of distinct linear factors, i.e. Q ( x ) = ( a 1 x + b 1 )( a 2 x + b 2 ) ··· ( a l x + b l ) , then (1) R ( x ) Q ( x ) = A 1 a 1 x + b 1 + ··· + A l a l x + b l Case II: Q is a product of linear factors some of which are repeated. Suppose ( a 1 x + b 1 ) is repeated j 1 times. Then the term A 1 a 1 x + b 1 in (1) is replaced by (2) A 1 , 1 ( a 1 x + b 1 ) + ··· + A 1 ,j 1 ( a 1 x + b 1 ) j 1 Case III: Q contains irreducible quadratic fac-tors, none of which is repeated. Then in ad-dition to the terms that appear in (1) and (2) the expression for R ( x ) Q ( x ) will contain a term of the form (3) Ax + B ax 2 + bx + c...
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