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# Class Note for MATH 1432 at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Lecture 10ection 85 Rational Functions Partial Fractions Jiwen He 1 Partial Fraction Decomposition 11 Rational Function Rational Function 0 Rational function 131 Where 131 and are polynomials 21 314 7 2012 17 1271727 1321277 12 1 7 lnz 131 7 NI 0 lfcle ree P 2 de ree 7thenb cl1V1s1on7 7 z g g Q y Qm 0 Qm Where 101 is a polynomial and is a proper rational function Le7 degreer lt degreeQ 7 Yes 7 No 12 21 1 1272173 1272173 13 3172 2 12721l 1 12721l 12 Partial Fraction Decomposition Partial Fraction Decomposition Partial Fraction Decomposition 0 Let the denominator factor as a 7 Oil7 HQQ jz 7139 Where the quadratic factors 12 jz7j are irreducible Le7 747139 lt 07 they have complex zeros Hm Qw can be Written as a sum of partial A BzC 171 12 zvk 0 Proper rational function 131 fractions of the form 7 Each power 1700 ofa linear factor 17a contributes 7 I 7 a I 7 a 7 Each power 12BI7n of an irreducible quadratic factor z2 z7 an On t b t B2IC2 BirCi con r1 u es EH9H7 In9H7 12 r7 Partial Fraction Decomposition Example A 0 Each power 1700 of a linear factor 17a contributes in I i a 0 Each power 12BI7n of an irreducible quadratic factor 12 z7 con t b t BnICn B2IC2 BirCi n u es mummy mu9H7 12 r7 Examples 1 QMi 1 1312l713 12 z z2l 713 z z2l lCDI4BEISACI2BIA A1 30 AC07 BE07 CD0 1417 B07 0717 E07 Dl 12 A BIC DIE zil12l27171 12l2 z2l a 12ADz47DEzg2ABD7Ez2 7B07DEzA707E AD07 7DE0 2ABD7El7 7BC7DEO A7C7E0 Finishit 13 AIB CID 1221227z22122 z2212 a 1301320D12A202DzB2D 017 20D07 A202D07 B2D0 Finishit 2 Integrals of Partial Fractions 21 Partial Fractions 839521 3 artial actions ions 700k A2 A dzAln zia C 17a A A 1 ziakd1iki1 17ak 10 Examples 2 6 6 13751261d171172173d1 7 1 3 2 dz 7 z 172 173 ln z 731n1172121n11731c 1 1 137212dI12172d1 lt712714 14 gt dag z z 172 7ltEiln z ln1172 gtc z dIm lt 14 112i7112gtd1 1732 173 z32 13 1 3 lt mln x3 3371n1131gtc 1 Integrals of Partial Fractions 3 Partial Fractions M B AL 22 rv 7 22 r7 7 B C B C B E 2Ldziln 12 xry 725tanilz72 1 51 7 2 7 7 7 4 v7 4 Proof dzidu1n u Cln z2 z7 C Setu12 z7du21 dzi 1 1 1 2 12BIVd1t2a2dta2seCQuasec udu 1 1 1 t 1 iduiiiuCtan71 7mn 1172 a a a a 32 32 7 4 7 4 Note I 61 v z 22 7 76241 Sett1 27a277524 2 set atanu t asec2udu dt t2 a2 a seCQu Integrals of Partial Fractions Examples B 0 7 B 2 H3 Cig Partlal Fractlons m 3 W B C B O 7 5 9 deiln z2 x 725tanilxi2 12 61 7 2 2 52 7 7 T 7 7 T E1ample 3 r2d1lt gt dag 11124 5 11 124 1lt1 14732 M 5 11 124 124 1 7 1 3lt1n11 21n1242tan 1 gtC 12512 dIlt gt d1 11121 11 121 1 21 3 ltmmmgt 1n 11 ln1213tanil1C 1 1 11 11211dIlt11211gtd1 1 1 21 1 1 77 d1 lt1 21211 21211gt 1 1 1 7 2 ln 1 1n1211 tan 1 C am BA WQMt a E Tummy 7 2 212WV 1 gwk B1C 7 B 1 M4 12 z7kd1772k112 z7k 1CCOS 7 W 1 1 1 2191 Wmdua2ki71cos udu Note z2 z7 z 2277 241 Set It er Q7 d2 776241 2 set dtanu t dsec2udu dt7 t2d2 d sec2u1 1 n 7 1 Reduction cos z dz 7 cos 1 z s1n z cos 2 z dz n n Integrals of Partial Fractions Examples 7 BziC g n1 Omit Partial Fractions 12443147076 7 2 12 17k 12 7k BzC 7 B 1 WM ltz2 z1k 2ltk 71gt z r7k 1 7 WhereciLg tiz 2 042 7524 dtanuit 7 12764 7 7 7 77 7 7 7 Ezample 4 3z4z320z23z31 2 z 1 dz 7 dz z1z242 z1 z24 z242 1 1 21n1z11 lnz247gcos2udu 1 1 21n1z11 lnz247EusinucosuC 1 2 1 ilz 2z 21n1z11 lnz 47Elttan 5477270470 Out line Contents 1 Partial Fraction Decomposition 111 Rational Function 1 Partial Fraction Decomposition 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 2 Integrals of Partial Fractions 2 1 Partial Fractions i i i i i i i i i i i i i i i i i i i i i i i i i i i

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