Method of Partial FractionsPartial Fraction DecompositionSlide 3The ProcessA VariationSlide 6What IfGotta Try ItSlide 9Even More ExcitingCombine the MethodsTry It This TimePartial Fractions for IntegrationWhy Are We Doing This?Assignment AWeierstrass SubstitutionWeierstrass Subs In for FizbaneAssignment BMethod of Partial FractionsLesson 7.4Partial Fraction Decomposition•Consider adding two algebraic fractions•Partial fraction decomposition reverses the process3 2?4 5x x- =- +23 2 234 5 20xx x x x+- =- + + -223 3 220 4 5xx x x x+= -+ - - +Partial Fraction Decomposition•Motivation for this processThe separate terms are easier to integrate223 3 220 4 5xdx dx dxx x x x+= -+ - - +� � �The Process•GivenWhere polynomial P(x) has degree < nP(r) ≠ 0•Then f(x) can be decomposed with this cascading form( )( )( )nP xf xx r=-( ) ( )1 22...nnAA Ax rx r x r+ + +-- -A Variation•Suppose rational function has distinct linear factors•Then we know( ) ( )23 1 3 11 1 1x xx x x- -=- - � +( ) ( )( ) ( )( ) ( )23 11 1 11 11 1x A Bx x xA x B xx x-= +- - +� + + � -=+ -A Variation•Now multiply through by the denominator to clear them from the equation•Let x = 1 and x = -1•Solve for A and B( ) ( )3 1 1 1x A x B x- = � + + � -What If•Single irreducible quadratic factorBut P(x) degree < 2m•Then cascading form is( )2( )( )mP xf xx s x t=+ �+( ) ( )1 1 2 2222 2...m mmA x BA x B A x Bx s x tx s x t x s x t++ ++ + ++ �++ �+ + �+Gotta Try It•Given•Then ( )3222 5( )2xf xx+=+( ) ( )( )( )( ) ( )32 222 23 23 22 522 2...2 5 22 2x Ax B Cx Dxx xx Ax B x Cx DAx Bx A C x B D+ + += +++ ++ = + � + + ++ + + + +Gotta Try It•Now equate corresponding coefficients on each side•Solve for A, B, C, and D( )( )( ) ( )3 23 22 5 22 2x Ax B x Cx DAx Bx A C x B D+ = + � + + ++ + + + +?( ) ( )32 222 22 522 2x Ax B Cx Dxx x+ + += +++ +Even More Exciting•When butP(x) and D(x) are polynomials with no common factorsD(x) ≠ 0 •Example( )( )( )P xf xD x=( )223( )1x xf xx x- +=� -Combine the Methods•Consider whereP(x), D(x) have no common factorsD(x) ≠ 0•Express as cascading functions of ( )( )( )P xf xD x=( )( )2andi k kn mA A x Bx rx s x t+-+ �+Try It This Time•Given•Now manipulate the expression to determine A, B, and C( )( )225 4( )1 3x xf xx x- -=+ -( )( )( )22 25 431 3 1x x Ax B Cxx x x- - += +-+ - +Partial Fractions for Integration•Use these principles for the following integrals( )4 2221x xdxx x- +� -�256 9xdxx x- +�Why Are We Doing This?•Remember, the whole idea is tomake the rational function easier to integrate( )( )( )22 222 25 431 3 12 1 11 32 1 11 1 3x x Ax B Cxx x xxx xxdxx x x- - += +-+ - ++= -+ -+ -+ + -�Assignment A•Lesson 7.4•Page 455•Exercises 1 – 25 oddWeierstrass Substitution•When using substitution forintegration involving sin x or cos x•For -π ≤ x ≤ π let 22 22tan then22 1sin and cos1 12then 1xuu ux xu udx duu=-= =+ +=+Weierstrass Subs In for Fizbane•Use the technique for the following sin cosdxx x-�4cos 5dxx +�Assignment B•Lesson 7.4B•Page 456•Exercises 27 – 53 odd (skip
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