MATH1014_integration_IV_Fall_2019_20 - Techniques of Integration Partial Fractions(Mainly based on Stewart Chapter 7 \u00a77.4 Edmund Chiang MATH1014

MATH1014_integration_IV_Fall_2019_20 - Techniques of...

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Techniques of Integration: Partial Fractions (Mainly based on Stewart: Chapter 7: § 7.4) Edmund Chiang MATH1014 October 8, 2019 1 Introduction Let f ( x ) = P ( x ) Q ( x ) where the P ( x ) and Q ( x ) are polynomials. We want to integrate Z f ( x ) dx = P ( x ) Q ( x ) dx. If deg P ( x ) > deg Q ( x ) , then the division algorithm allows us to write P ( x ) Q ( x ) = S ( x ) Q ( x ) + R ( x ) Q ( x ) = S ( x ) + R ( x ) Q ( x ) , where S ( x ) and R ( x ) are polynomials and deg R ( x ) < deg Q ( x ) . Example . Decompose Z x 3 + x x - 1 dx Since x 3 + x x - 1 = ( x 2 + x + 2)( x - 1) + 2 x - 1 = ( x 2 + x + 2) + 2 x - 1 . 1
C.2Partial Fractions type ITheorem (Partial Fractions I): Suppose all the roots of a poly-nomialQ(x)are simple roots (each root appears once), that is,Q(x) = (a1x+b1)(a2x+b2)· · ·(anx+bn). 2

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