Integration using partial fractions

Transcription

1 MTHS 50: pre-semester study Itegrtio usig prtil frtios This tehique is eeded for itegrds whih re rtiol futios, tht is, they re the quotiet of two polyomils We sometimes use log divisio i order to rewrite suh itegrd ito sum of futios whose tiderivtives we esily fid Rell If p is polyomil i the vrile, the degree of p, degp) is defied to e the highest power of i p) Emples hs degree, hs degree, 5 hs degree 0 We ould write this: deg ), deg ), deg5) 0 Revisio of log divisio Emple Simplify usig log divisio We divide y ) We re goig to deompose the umertor ito produts of the deomitor, where possile The followig whos some steps of the proess, egiig y writig ) ) We ow sk, how my times will, the term with the highest power of the vrile i the deomitor,, "go ito", the term with the highest power of the vrile i the umertor, Sie, we sy it "goes" times d write o the lie ove ) ) ) Result: ) ) ) 0 Emple with remider Simplify Now multiply: ), write this elow, d sutrt this produt of the deomitor from to give remider, term tht hs ot yet ee divided y the deomitor e very reful to use rkets here Look et t the term with the highest power of i the remider,, d sk, how my times will "go ito"? Sie, we write o the lie ove Now multiply: ), write this elow, d sutrt this produt of the deomitor from The proess goes o util we hve zero remider, whih must hppe i this se s ) is ftor of,, d it is ow esy to fid tiderivtive usig log divisio

2 MTHS 50: pre-semester study ) 6) Divide first y The remider is : we ot use log divisio to divide this y euse the degree of the deomitor whih is ) is higher th the degree of the umertor ) hs ot ee divided y ) 6) We reogize this y simply writig Result: ove Emple ) Do it yourself Prtil frtios If the deomitor of rtiol futio is ot simple lier or qudrti polyomil, s i, the result fter log divisio will ot eessrily e sums of futios whose tiderivtives we esily fid The tehique of prtil frtios is method of deomposig rtiol futios, d is very useful for preprig suh futios for itegrtio d hs my other uses lso) osider, we esily dd y fidig ommo deomitor ) ) 5 ) ) ) ) Wht we would like to do is the sme thig kwrds, euse the right hd versio is ot somethig we would re to itegrte, while the left hd versio is perfetly resole Defiitio The qudrti polyomil q give y q) with oeffiiets,, IR) is sid to e irreduile if < 0, s it ot the e rewritte s the produt of two lier polyomils with rel oeffiiets This is just usig the qudrti formul to fid tht if < 0, the the equtio 0 hs oly omple solutios, d so, y the ftor theorem whih sys tht pd) 0, where p is polyomil if, d oly if, d) is ftor of p), hs oly omple lier ftors Emple,, re ll irreduile Method of prtil frtio epsio of rtiol futios p0 ) Give where p 0 d q re polyomils for whih degp 0 ) degq), we use log divisio to rewrite q ) p ) the epressio Oe we hve epressio for whih degp) < degq), we my rewrite p ) q ) q ) s sum of terms lled prtil frtios, whose tiderivtives re kow I order to do so, we first

3 MTHS 50: pre-semester study osider the ftors i the deomitor We sy the ftor ) or ) of q) is repeted times where ) or ) is ftor of q) Emples: ) hs repeted twie d repeted oe i the deomitor ) ) hs repeted three times d irreduile qudrti) repeted twie i the deomitor Kowig this, we ftor the deomitor d the write dow the prtil frtio sum or epsio), usig ukow ostts The sum of prtil frtios iludes see emples elow): the terms ) ) for eh times repeted lier ftor ) i q ), where the umertors,,, re ostts; the terms ) ) for eh times repeted irreduile qudrti ftor ) i q ), where the umertors,, re lier Emple d The itegrd is rtiol futio quotiet of two polyomils) with degree of the umertor less th the degree of the deomitor, s 0 < We my use the method of prtil frtios to deompose the itegrd Step Rewrite y ftorig the deomitor, d mke the required ssumptio: ) ) ), ostt divided y lier term for eh lier term i the origil deomitor) Emples ) ) ) ) ) ) ) ) )

4 MTHS 50: pre-semester study Step Fid the vlues of d for whih ) is true for ll There re my methods, we will use two of these, d oth require us to first multiply oth sides of ) y the ommo deomitor to get epressio without frtios ) ) ) ) ) Method : the over up rule The over up rule is sed o the ft tht if ) is true for ll rel, it must e true for y prtiulr So we hoose vlue of whih mkes oe or more terms o the right i ) zero d we reple this vlue i the equtio I this se Step replig i ) gives 0 ) replig i ) gives ) 0 if ) is true, the d ) ) ) You lwys hek your result y ddig the frtios: of ourse you should get k the origil rtiol epressio Now we my itegrte: d d [ l l ] l l ) l ) l ) You my fid you ru out of vlues for tht give zeros i ) efore you hve foud ll the oeffiiets I this se just hoose other simple vlues for tht hve't yet ee used You will ofte get some simulteous equtios to solve We eed further result efore we use the seod method Lemm: If y two polyomils hve the sme vlues for ll IR, the the polyomils re idetil d so the oeffiiets of the orrespodig powers of i the two polyomils re equl i This mes tht if 0 0 i is true for ll IR, where i, the i i ) ) ) 0 for ll IR d 0 0 i i i 0 0,,, i i d i 0 Emple We will do the lst prolem gi usig the seod method The proedure follows the lst oe up to ) d further epds the right hd side to rewrite it s the sum of powers of ) ) ) ) ) ) Method : equtig oeffiiets y the lemm ove, the oeffiiets of the orrespodig powers of i ) must e equl s, the ) ) Step 0 : where we equte the oeffiiets of 0, tht is, the ostts, eh side of the equlity : 0 where we equte the oeffiiets of The oeffiiet of t left is 0

5 MTHS 50: pre-semester study We get the two equtios 0 d, d solvig this lier system gives us d s efore Whih method is est where? The over up rule is quikest where the deomitors i the prtil frtio epsio hve lier ftors Equtig oeffiiets is usully etter where the deomitors oti irreduile qudrti ftors Emple ) d Step ssumptio ) ) with ostt umertor for the lier term d lier umertor for the irreduile qudrti term Step Multiply this equlity through y the left hd side deomitor, d euse we hve irreduile qudrti term, epd the result s sum of powers of i order to use the seod method of fidig the ostts ) ) d therefore ) ) ) ) Step Equtig oeffiiets i ): : 0 : 0 : givig,,, so tht ), The ) d d To itegrte, we will split this itegrd ito three prts, where the first is ovious split The reso for the seod my ot e ovious immeditely Look t the deomitor of the seod term : we would wt to mke the sustitutio u, whih eeds multiple of to pper i the umertor if it is to work However oly oe prt of the umertor ) is multiple of, the other is ostt, so we split the quotiet i order to del with eh prt seprtely d l l t ) Do this itegrtio yourself, d ote tht we hve used the followig list of tiderivtives whih you ought to kow y hert For, ostt, si ) os ) t ) os ) d si ) d se ) d e f ) e d d l d l f ) f ) d t ) 5

6 MTHS 50: pre-semester study Repeted ftors i the deomitor Where there re repeted ftors, tht is, powers of ftors i the deomitor, we must reogise this i our deompositio ssumptio Emple Evlute d ) First hek tht the degree of the umertor of the itegrd is lower th tht of the deomitor If ot, log divisio must e doe first) The write dow the deompositio ssumptio ) where euse ws wht we ll repeted ftor i the deomitor t left, we eed to ilude the terms with deomitors d eh with ostt umertors, s is lier) Red pge gi The, multiplyig through y the left hd deomitor gives ) ) Usig the over up rule, we look for vlues of tht give zeroes t right : 0: Hvig ru out of zeroes efore fidig, we use y other simple vlue of : 8 givig The ), so tht d d l l ) l) l) 0 l) ) l) Emple Write dow the ssumptio for the prtil frtio deompositio of ) ) First hek tht the degree of the umertor ) is lower th tht of the deomitor 7) The ote: ) hs power, we eed repets util ll powers re used) with ostts i the umertors s it is lier term ) hs power, we eed repets with lier terms i the umertors s it is irreduile qudrti ) ) ) ) ) We would the solve for,,,,, d I suggest y the method of equtig oeffiiets) However it is it sty, so do't tully do it Itegrtig the result is lso sty Emple hek degrees Fid prtil frtio epsio of 0 5, d hee fid ) ) 0 ) 5 ) d ssume tht 0 ) 5 ) multiply y LH deomitor 6

7 MTHS 50: pre-semester study 0 5 ) ) ) d sort the RHS y powers of 0 5 ) ) ) Equtig oeffiiets: : 0 : 0 : 5 We get system of lier equtios i, d : givig,, 0 5 Therefore ), so tht 0 ) 5 ) d ) d 0 7 To itegrte the result, we eed to split the seod term, euse the i its umertor mkes for ie sustitutio, ut the ostt eeds differet tretmet: d d d d I I I l Usig oe of the shortuts, I l l ) ) du du l u l I eeds the sustitutio u d I u l ) s is lwys positive, this e rewritte to I hs itegrd whose tiderivtive is iverse t futio We rewrite it to I d d d Now mke the sustitutio ) u du d du d use the formul give i u t u ) the tle o pge 5 t l ) Therefore d l t Redig: D 85 Eerises: D p 5 #, 5,,, 7, 7 7