L16 - Integrating rational functions — 1 Problem Evaluate...

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Unformatted text preview: Integrating rational functions — 1 Problem. Evaluate / m dx. NOW ‘oniC ’be miﬁomk QMXM DL'\ be, mwﬁ’t’ren m5 “:5;va I ")_ m6} (Ni-pa U\\\3 A a /\ (Chad’- ’Ms‘) ”“3?- K'bewa5 “ii—g 1-7, 7" \$ 1’ 2 “\mihcﬁ gmL‘Mtxs We, cam W05 vomit " .- "‘ "'___—d AX- : _—-—:—- ——-‘——” jlz*:1+é AOL - J1-’5 DC‘Z. ‘ 9L 3 Mp3“ w 1-3 SD A“ \$379 ‘ w ’M 6 Sets m - , , A”. ﬁl< “‘3‘ l S?- . F - - ’50 chi- \\“ t: glean“; E1053 ; 0 Seﬂﬁe, SOM‘NIWM \l , 1. 7.. “M5 .— '2‘ Arc W Av The general problem of integrating rational functions P (X) G(x) If degP 2 deg G, then (by long division) there are polynomials q(x) and r(x) such that P (X) 7‘06) G(x) G(x) and either r(x) is identically O, or degr < deg G. The polynomial q is the quotient and r the remainder produced by the long division process. Problem. Evaluate / dx, where P and G are polynomials. = 6100+ If r(x) 1s 0, then (T3? is really just a polynomial, so we ignore that case. Now,/G—((:; dx=/q(x) )dx+/ G((xx))cbc. We can easily integrate q, so the general problem reduces to the problem of integrating a rational function (—05; )) when degr < deg G. P So, suppose f (x) = E with degP < deg Q. QCx) \$335?” Q can be factored as a product of linear factors (i.e., of the form dx + 6) and/or irreducible quadratic factors” (i.e., of the formax2 + bx + c, where b2 — 4616 < 0) . Our strategy to integrate the rational function f (x) is as follows: > Factor Q(x) into its linear and irreducible quadratic factors. > Write f (x) as a sum of partial fractions, where each fraction in the sum is either of the form K _. w or Lx+M (dx—l—e)S ‘(axz +bx+c)’ > Integrate each partial fraction in the sum. Question: how do we ﬁnd K, L, and M for each fraction? Let’s look at some examples. Examples 4x2—3x—4 E 1 LE 1 t ___.__d _ Kampe vauae/x3+x2_2x x |( WE‘LJP‘G- “N am: 13*“1'511 = 1(11'4-1-93 = Mac—Nani), 60 at») has “me— AD‘W‘J‘ “e“ Sime, each guﬁor a??eo«r‘.’> exoxdﬁg aﬁce.’ we, Cam wd’ve’ HoLla3x-Lk “ 133+ E) +~—C:-—- gr Some, “widows Algomch. 1134—11—22. ‘— X 14 1+2. ‘ ’ocw) 5 To \$06, ’hese, nombeYs) M's m0\3r{(>\.9 Snow’s sides b3 1(1— H172 son-Dc 7— Mob—060:2) + 3:414:13 + C341") ' H a le’i's choose, 052M wakes of at. 1 NM NS eqon‘h'un 15 ere, for 9.. x, ) 5 9.31:0 => ——‘-\ :raA =5? #:9— 191 z) «3-; 355>9>='\ lrl ==>1Pp=éC=>C”3 “mm.“ Al :_ 53;, ”2.43.1. Ax, - aMM \ f+£~2x 7" 14 1+ 3—4 —1 Example 2. Evaluate / x__x___ dx. x(x — 1)3 5 .- .m _ Hem C1003: 1(1-1) 501m, “MM 55hr (1—1) 6 womb} 5 41 e5 ) \n *‘MS Case, om” Pwkm gmddans we 0‘? ﬁst—1‘1”“; D ﬂaunt.“ ’_ j: 4— ...-E->— + C 1. +(ﬂfx-1)?’ g“ Some, nombe 3L( 03’ " at 1—1 (1—1) x. ’c “\ ‘0 QC“) ’m 33 Pm Magma, mo\5«\p~3 3 ’5 ( 31+ C3414) +DL -« - {Mac-Q + Ebacx 3 _— 1~H1—\ 9‘5ch choose, 0508A m\v~es 9N x, “5' 1:0 .3 424% => AilI—k 1:1 :4. D in). 4:) B+C=3 5°, ‘ ’\ ‘ +26+zzc+29 ._ mane - 3 ng .9 — .- Pf - «H©+QC-H=—1B’rt— 1:4 9 3:»%A*H6+ZL*D - {5 Aﬁbi C, aunéb. ’“wS 3 1 ' BL' '1 ‘ 3’ ——L-L— 67L =- jMWW ’ 1 +-"“ﬂz+[ 8:; 1|? A1, 2 1— &4? (”D3 1' (1* 1" 5x3—3x2+2x—1 Exam 133. E 1 t /_______ dx. p va uae x4+x2 Hm, QM): 36+???- llblﬂ) ,50 Q00 hob onﬂmewf ‘FN—JKW % WYéa’ieA Mice, and one, disSx'mc’: Ewe Audbk/ Q’Utké m’sfc, ﬁxd-or 114—1, Our 99.86“ ‘gYoéxx'onS ME, 0? We, {arm ; . 5x3v51L+ 2x-\ 3.“ ML” (5 Cami) 1;“ maﬁa“ A)€>3CM3(D 4.. 1" 11+ 1 _ .9. + ‘— 7L “\Nﬁeﬁmﬂ 53 (31(1) , v06 36‘? 59L5~3f+21~1 = M130 4- 606+") + (MM =(mc)13+(a+n>mMB Tc“? 'Hme. we. Com Lum‘wd’e. Co—egaden‘hﬁo see TWA" 31-1 )Ai 2- , 5‘”):- “'5 ma A+C:5_ .. ) ”was; (a, Pr—l, Lav-«Jazz; omA «pz—a. we, cm nova mum We, Marni: 1' -- ‘lfaﬁ‘()‘+' j?" ‘ +51”?- Ax, : £1ﬂx~\1\+3£+%9w\1“\ I). 1' K” .— .— 3513-31L*21"\ d '1. 1H4, 12. .1 ‘ W“ K 5 2' So U591 0" ”bah“ 3’5“ 1741 11+! ) u' 1"“ 5“)” I (“hard . dx Exam le 4. Evaluat / ————-—--. p e x(x2 -|- 1)2 HQ“! QM) 1 101111)?- hus one, \Rnem" WM 9L, 0W3 one» cLuaérah'c 9“)?" 78-” YEWOA'QA “we“ 1H5 means ﬁe, Vowsﬁcﬂ 'Qrach‘oms \oo\t \ike, ’foIS: ____.‘.._—- : 31+ 01:33) + Efﬂ: 5f conﬁm’f’: ALT); maF 7L 1,4" (1+0 I(1"+ 1)1' mwéﬁwj ”’3‘“ 513‘“ bf) 1&5”? 33u2505 \ ' A(1"+15L+ (Cx+D)9L(1‘+I) + (Ex+F>9L 2: (M639? + 913+(ZMQQx - - +LD+F37L +A 0 D-VFT-O own! A:’\, I ' ‘ ‘ I ‘- 10 C—HE : ("Balm1 LOMYMHB Lo-egguevxk'b gwes 05 MC. 0119 ) 2A): A;’\ C:—’\,D:OJE:—\ MA?:O. ’“nox’r {5, SO) \ M\ 11*“ _____}___ + \4 iii- :. .3— " “BE—— "' L1 AOL r. ”VIA —- :2" 3L 2&1“ j 1W“? 9“ 15 I (11+ 1) 9‘5““ use, sobe’fi‘tvﬁm 1 12M hora: {L51 Example 5. Find the volume of the solid obtained by revolving the region R x — 9 x2 — 3x between the curve y = and the x-axis over the interval 1 g x g 2, about the y—axis. 0%an memo} o? gnaw,’ We, \jo\UMQ_. We WOWT‘ 18 I. 2. z 7.. q " IUT 1.310141. :. lﬁgiﬂiéoc. _—_ QWSLg—p 11-51 ‘ Mac-8) \ 16 t \ We, (amok enméﬂn‘DSé-I Z a, in mm o8 0? WW?“ 3211'] - E 49‘— {YO‘AiUnS’ bo)‘ Pom 6 I wwriieibiﬁ m a” 7— uscgﬂ ﬁrm- The steps to integrate a rational function f: a technical look Suppose f (x) 2: % with degP < deg Q. 1. First factor Q(x) into its linear and irreducible quadratic pieces. If there are n distinct linear factors and m distinct quadratic factors, then Q(x) = (dpc—l—eﬂsl ...(dnx—I—en)s”(a1x2+b1x+cl)t1...(amx2+bmx—l—cm)tm 2. Then f (x) can be written as a sum of partial fractions as follows: K 1,1 K12 K 1m x = +———+.. + f( ) d1x+el (d1x+el)2 (d1x+el)51 K111 K112 K713 + ’ +—’-———+...+——i—-—+ dnx + en (dnx + en)2 (dnx + en)“n L x—l—M L x-l—M L x+M + 12,1 1,1 + 5,2 1,2 2+.”+ 12,11 1,r1 t + alx +b1x+01 (alx +b1x+cl) (alx +b1x+01)1 + Lm,1x+Mm,l + Lm,2x+Mm,2 2+ + Lm,tmx+Mm,tm amx2 + bmx + cm (1:1me + bmx + cm) 0 I I ((1me + bmx + Cm)’m ...
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