Calculus Third Editon By Strauss, Bradley and Smith sec7.2,7.3

Calculus (3rd Edition)

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Unformatted text preview: Section 3'2. fntqgratfon by Parts 7.2 Iangrntion by Paris. Pagan 439-411 1. 2. = -%rc‘“+%[u_h d2: = ‘EIT-E — -:-c'2'+ 0 [mm —:c.os:+ Jcmzd: = —:wsx+sin=+6 junm = %a3|nx — 'E'Illdl‘u =1r2|I1x—%LR+C [sin-1m =xain_’r—I M“ 31—13 =rnin_1x+Vl—-r2+C [an-am: = —z=cm+ajmm= : —12cna r-I-‘J{.I:ain a: - Jainrd'x) = —:.3coa:+2::ain:+2coar+0 Page 31.? 7. I= Ie‘hmdrdx 1. ll 2—3'sin 42+ file-3‘ sin 41: d: =%e_3‘ain II: — figs—s‘m‘h] - fijc‘a‘cw 4: d: §I=§c'3'sin4r — fie'azmséz+ Cl I=fic-3'sin 4: — %e_um54:+ C a. I= [era-mam —§c"m 3:: + H‘sch 3:: dz: Je’rm a: + gym“ 3: 41.2%“: 3:.- dz] I1 = 432.: 3: + yam fix— 3H :91 1,34: -.}¢"m 32+ gehaiu sz+ C1 = -1?§ez‘ms 3r+135ehain Ez-i- C n. Imam: =§¢'ln:-§[=’dz=§fluz _ $9“: 10. It: +,sin gin: [(23+2nin z+ain“=) dz =§23+Elxsinxdr+ Iain“: dz =H+2isin :— xcoafl+{§_flin42rj + c = g + k+2fint-2rmru%sin2: + C 11. 3} Jr: “man :3 '— [mafia 3:} a: = “infill I) - mfln r] — Iaiufln I] d: Page 313 Chapter .7. Mcthod's of Integration 2jsinan r) or: = 451an x: — mun =}1+ 6. = [mun 1'} dz = asinfln :) — cosfln 2]] + C = In it: 12. [main mmrdrz lilsain 2:: If: =1'Iu2 + C i =5-[—%zcoa2r~J.—'§msfirdr] =éluw]”+C _ I I l - _ —ix cm 2r+ :6 am 2.!) + C :%1n2(51n I] + C =—i~rcos‘2:+}l{sin2x+c ‘t 17 Ifilnrd: uzlnndv=fid 13. [Infill-F I} d: u: In“:2 +1}; dbl: If: 1 4 ‘1 4. 2 12 2:2 =22‘3'llnz'iI —3§J.::'"r d: =$lll(l-'2+lj— F—dz 1 +1 _16 43124_32 _2a _ — In 4 — —z _ In 2 I a 1: 1 T "g" = £1.1qu +1) — 2K1 - f) a: ' + 1 Calculator value: 1.282413 2 —1 =x1n{:2+1]-2:+2t.am I+C 13- [Phlsz "=lnndfl:x3d 1 e . 2 _ '— 14- [atnfidz = if“. 3‘: _ fix: a, 1 :J-ainuflwdw] =icq_1lgx4|:=1:n4_%ggq+fi = zjwsm w w flag + 1) : 2{—- mam w+ [can :1: dun] Calculator value: 16.29965 8 2 =2[_wmsw+sin wl+c 19. Jfllnx} d: u lnr: u r 1 c :_?’ Imflfi-l—hinfi'lc : mfllnzjzlf-Zflnrdz 1 u=||1::;tl=d. c = a|[|n a}? _ 21:11:: I": - [dz] _ 1 =c-%¢lnc-llnl-rlf]=e—-2 Calculator value: {1.71823 8 an. 12.0mm: lan 1: 13:: 3|! = (law: 8 l Section 312. integration by Part: Page 319 3n = t[lnt)2F:—2I1ntdt [MED—dds t:ln.a:;dt=.r_1d. l = Imam ‘3 =a1m6i+lcofltdi=-teost+aini+c = 31201133]: — tin 1P1“ # Id: 1 = —]n .tcoafln r] +sin(1n z) + C _ 2 _ _ e ' sea“ 3 +1} Eiacfin 3c} 1?1.] 25. I[fin2:h(m:]]dx:[2 sin rmaxlulfieoa 1) d1: = semis. + 21.13“) r2[35{ln3+1]-3e+1] ‘=‘°°“i'=‘“'“ =a=m=a+3h2 =_2[unm Calculator value: 15.913733 1 = —{;Fln1—iltdi] ;= J 2m ‘1' ,1 = —t’ln Hit” + G =‘5e2‘sin ital: —- Jehain 2.1: cl: = arenas: — [meat] ln(eoa 2:) + C E! 25. Ié‘fine‘&:=j¢'sine’[e‘dx) — 24: 1T , [-1,2xm,lg+[etm2m] = [want I " =—1mt+lmtdt:—ieoat+aint+fl : ifltfii-ir(1)(fl)+%c“(1]4i(lltll — r I 1 =-t"ma:‘+aine"+c _ ESI' 2f— EC — E 27. [[finxh(2+m:]] dz: — Jim I fit I w 22. Jriainx-lvwar} dz :1 = a: tie: (sin x+eoa :] u = —llnt+£+C = —[2+cna z}lu(2+coa z} + [2 +1105 2:] -+- C 11:31.11“: rd: :3 25" i?—1d‘ ?—1 saw — “tittie— mt- : as?" - i] Calculator value: 133.62291 :r :ayf—cusa:+ain 3H —J(—cos :+aia:} d: a =1r+(ai|tr+eoa tug: :- - 2 Calculator value; 1.11159 :3. Integration by parts is the application of the formula. = ,3}:an - ll -—'EJ1111::2 - 1|(2: dz] l””“"' i”“ =t£luiE-1l—§{E—1)Inir’—1l The H factor is a part of the integrand that is + a? — l) + Cl differentiated and tin in the part that is _ integrated. Generally. pick file as complicated — tux-i=9 - 1L+ :2} + c as pouaible yet still. integrable, so that the :3 integral on the right is easier to integrate b. cl: = ID: + :(122 - 1.] ’1] d: than the original integral. Chapter 3'. Methods of Integration Page 320 #1 1 _ 1“” _ :“+‘ _ 3.3+:E11'IIEFII‘I'C —mln= =%(lnl:1 — 1I+32I+C 34. Let fit) he the distance traveled. 29. ape”: df= 1r”? 1": -_.I | 1__, n. 0 I 1:. 53 Flu H. = sin 1:03 z+Jisinzx #1: I1 I In: R I. .2. W + b: "——'| G 1 II I DO .k H. 1 -T. u I .:h. n I n. 1-. n + C} =sin zoos x+ Ifl —ma=r]d: :sin .rcus 3+ :— [mash = -2e—'”(2 +1] + c 21=ain zmar+ 1+ 6' I 35. - Let (fill) be the number of units produced. _ #3.“ I _ laflflte I=%{x+ain=coax]+c it? :I" E du=dqu=i+gfla£fi¢umhIm29 =-lflfll}e_‘l‘5+liflflnslTTunits 35. Let (2U!) be the amount of money raised in t —_- + 5112:] __ + 311142;) a "Rh- £Q_ FUJI =1§+%xsin2:—&§+%m2:+c W‘2;Dflmc = 31:! + w“ 2” imam a can) [zoom :1: D : 2,nun{—5tc"”-“] l: + 10,000 I I: “ma =[—1u.uuotc‘°-“ J 50.0%?”‘2’H: x3 if: I: :12. I 11::2 in: I: du:2zdz,u=y‘?+l[roumblflnlfl = 409.0%" + 50,000 a: $13,212 VF+1vI2IV9+1£= 2 ar- v=2wj=r=n viii-l El 2 _:2 .1 33. [mm W "(‘" “3*? “J Section 12% fntegratr'an by Part: Page 321 2 21(1 — 3:: ‘2] 5:: 3.73213 41. The curves intersect when .1: = I]. Assume the up: density is 1. 1 3-8- V=21J zflainz+cmsafldt A=J(c,_e_,}dz a u . . W"- = 2111-5203: + ran-13+ smx+ «Ist a =2rrw'i — I] x 2.6026 33:; {c' F c")[a‘+¢“}dz c 39. a. VzurJIUnflgd: e = nfln fill: — EIlen ’3 1'33“ 2 1 I: My: [gif— e‘") d: : {1:{In — 27r(:]n .1: — rjlll a d = (i = — _ =1rl:e-'2¢+25—-2) 5‘ x; u a a] = «(a -— 2] #5 2.2555 =:(e'+a_']|;- [(cr-l-c—‘jd: c I: __ U _ 1 = —- g: _ h. V=21rlzlntdz=21r[5:-[2lnr—ml: HE +c I) I: '3 AIM-J ' {Sue Problem 3] =% a 0.7353 _ e2 1 _Ir 2 _ mtg-+3] -§(e +1) r: 13.1715 cm) W n: (0.63. 1.27] an. The curves intersect when a: = :rrfvi; assume the dcnsiLy '13 1. £1: _ ,_, _ “1' m ‘3‘ 33-“ “:1 . . I'M A = [{coa x—sm this: {smat+cnazjl n J'B-’ I" = Ire—W.” a = 2 — l 5': H.414? mm M =J%(coex-sina:][cosz+sinr)d'r U ‘:fl' "‘1 d w" Iy—uj dy: zuzlnxdz =% cos 22d: = %sin2s:| D = i— = [1.25 era a ff! My:[z{coax-3inz] d: u u=gdu=(coar—ninr]d 2y‘f1=§zmln=—gz3”+c inf-I 1.11:2 ['21 _ I C : :{sint+cus£}|fi:- I {siux-lnmag) d; 29 51:3 (11 .1: 5] + a m 44- £=Itan"a1noley:>o =33! 2 — (-casl:+ain:]| u [9—1 dy=Itau_1xdx =§ 2 —1 a 0.1107 In y=a~um":: — 1§ln(x2+1}+ C _ flrlll]? 0.25 (E, y} a: m. ‘3 [0.27. 0-50) : rta or )! Formuia. 457 Page 322 _£ _I|.12 Atlil,1]. 0—4 T+C _.1 _E Inn y: flan-11: -%1n{272+1:|+%|l12 —% y = exp[rtan_1:—%Ill[z2+l}+%ln2 — {L d9 fly 45+ -Jy" dy: Inward: By parts or - Fol-mu a. 312 -ln|yt=cuaz+rsinz+0 Atflfl.1).0=l+c,eofl= —1. —|qu1=wer+:einx —l y=exp{1—- one: — xfiin 3} 1r dli. + f"(::}] Bin 1: d: “(1') sin a: If: IJ ‘31' 11' = sinxdz+ If I] I] 1r = —eos zflrjlz-l— If'l’g) cos: dz 0 +sinrf'fix} q: — [fix] ward: a = —cm 11‘ + cos 01(0) This is 0, so that = —fl:|]] = —3. 47, um = —r1nw—;—fl— gt aft} = In“) d1: — £922 + l‘l‘. Since em) = 1] [the rocket. stem at ground level}, we find (by substituting the give information), {120] _ smuoymme- 200:12]]In(3u.M0—m[12n]) — T W. -1§{:«:2}(120)2 + 5.000(120} # 343.33511 9.: $5 miles Chapter 3', Methods offntegration sun—5]. 43. ——r1-—1[ mlagler som— —m' 3'5 "1—12 :5 TENS db 4!]. Average displacement is: arfs +1 2.3.: ' 0351015 5: e: 3 o E. e_ “51 41.25 cos 5: + 5 sin 51) (— {3.25) + (5} a: 0.0? 50. Let r be the distance from the camera to truck. Then 6 : ten-1(flr_]] — tan—1(e'1] The average value in 20 mfi-Ilm' 1w") Aerie-‘11 a: 4 a: HANS-15 (by calculator} The average value of fl is approximately I1. IQVI 51. int cWok: ljnRTln V “3 ' my 1?: luvl 3 :-.. “NJ (In v J In mm “1 V I 133* parts or y Form“ a 499 101? 1%.an V— V— Vln v11| ‘ 1 V 1 = H-‘glfiflin 10 — 9} fl es 1.553nET 52. Section 12. fnrggratr'on by Parts 53. [Her dz 1: = 5 5 . I": rhu = e‘ (If = zne’ — "Jr" _ 12" Jr -'1. [Ell] I)" d: H- : [In 1]"; 1h: = dz; = rfln at)" — :aJ.l_']I1r}“'"'d'r 11',” r: I sin“: 4h; n is even: '3 545- 1: : [fiin z)“_ I; (h! = sin I . fl H2 = -:;|n" IEHHII N2 + I {u - L}[:niu 2:)” “gnu-121' If: u w: = I] + (u - [:1]. [{sin z}“_2{l - ail|2r1]dr U I,” = [n — l}J‘ Lain 1'}"_1 uh: — [n - I}! U 17;“: "I = [H — HI {sin min—"5 rh- U Tr,” I: H—‘I [sianN "1 rt; 0 This rcuursiw I'nrlnuln will In: with! ritlillilhfilly wiLlI 'Ili'LlllL'fi. ul' H lel. IIUL'J'L‘ilhI: I9} '3. rm r=u+“—fi—1[5m“ fun- LI _n—1r5irL“‘3rcmr I”: _ T a 1']? 1'1— lu—J n—I: +--F[—"_2'[sm 1d: 0 1T}: _(fl-{II-Ii] ‘ N_4. _W 5m Li: I: 1: 1 if” _ _ 1-1—1 Lr;-3]-~Ll$ . _ _. SHIEI dz: PJSEJPS II (1m [Ii-IMH—SH-HJL _ q _ ' WJ{1+LEH-IJ¢E1 I] _ I':I-5-...{“_:§:Il:"_l]-£ If"! For I can”: dx‘ proceed as :ihown. album. [I 1112 I: J :03“: Jr, n is UIIIJ I] u — | ' I” = CH” 1' 5|” J--| U I}: -|— J [H - l]{:'.:m fl" - “Hing:- (If U 1'er U + (u. — IJJ [{cna r}“_ 1“ —r.nH21-.-]] Jr.- u If? = {n - HI {LL35 u:j"_1rh:—(u— [H II If? H! = {u —- [lil‘ {:er J:]"_": rl'J'.‘ u w: I: -—": II {cmsmff'fl2 d: U 'I'llin. munrqiw: I'urInLIL'n will In: Ilhtfll rupueuLcully with values 0|" H Um! ttliiH'IJI'I-‘rl: hr 3. if]! I=U+j—"_l|"[ ain't-ltd: fl _ !h— I can" '32: em 2] m " r1 -— 2 u w: + n _ 1 ms”_ 1: r11 D Inf? : cos" _‘lz' «11 .. "(u—'2} ‘ u an“? n {II-lli'i-33'-'[2} :...._ "(Hanan-(3} [coszdz' ll Page 324 I Chapter 3'. Methods of Integrativ- _2*4'fl""ifl—3l["—1} __t 3 15 " 13 5 ---[n—-2]n we“ ‘5“ +0 «I: For I sin": dar, proceed as shown above. = akin": — %slfl5£ + C CI 3. Jamar was: dz = sins: cost: (can 7: dz] 7.3 Trigonometri: Methods. Page 4-1? 3 = r — ain‘r’zflmt u: [sin 1. Convert Elna-1: = a]. — our: 22) and t: a _ a _ ‘ I. c0521“ = a1 + ma 2:) and substitute into the (H ) d" m given integral. = L—ein": — Janina: + C 2. Peel oil“ a factor of eecia: from the integrand and then nee the fundamental Identity 9. Ia.wa t sin t [it = Iuwfi — d'u] [IRE-Z": sen}: = tan“: + 1 to write the integrand in powers of ten 1:, except for the 994:2: ii: that = —§-n3"‘2 + C wee peeled off. Substitute u = tan n. = —2[coe 1}“? + C 3. Substitute u = a tan a and then integrate the 5 reeultmg Integral 1n trigonometne form. It 10- J‘lmB I a: : it; may help to draw a reference triangle. + ein : 4. Substitute u = a sin 6 and then integrate the = ‘élnl 1 + 35in rE + 0‘ resulting integral in trigonometric form. It may help to draw a. reference triangle. For 11. Jam" fight 2: dz = Ia“ [—du} V'ui - a“, substitute 1.: = a new. 5. Imam-Ila: Imixfimrdr) —e°°"+ C 12. {cougar} d: [in + cos 4:) d: z [[1 — n2][du] :1 r+§sin4ay + c =u-1'3E-FC =.}:+§sin4:+c = sin 1: -— §sin5=+ C 13. Jami: m3: d: 1. fl] —ene 2:][1+cne 5. Iain”: a..- = Jammie : £2] = [(1— fit: - dc) rm _I:E 2E3 _ *E‘l‘T I+C 1 ll 2 —%c055$+ 3:933: - Q06 2+ C 14- Iain 2 dz Iu—fi I _du) a... T. J‘einzx cm;3 dz: = Iain“: can“: (me. tin] = §lfcoe x] _" + G' 15. Jtenflfidfi' = —1|nlcon2fll+C 16. d3: = 21nlee¢§+ tan gl-i- C 1?. Items: em": d: = [mammaxfimgs ds} = [(sinlz - ain‘tflwe 1‘. it] =jM—«fin a3 Section 13, Trigonometrfc Method: Page 325 {max + Ham-,2: dz: : Itan3dtal1zm+lflacczzdrj = J = [um 1) in Mercl=lms 3- =-'§-+ n+ c 1 a l 4 =Ttanat+ tan r+C = Elm 3+ Elan 11+ 0 15. Inert”: tam 2 d1: = Iam‘flsm run 3: d1] 23. I at“ zone: d: = Julia-in _I_ 5 _. 59M: 2+ C m :gfl-fln m]4f3+ C 19. [Hangar-Ii 82122:] Jr: “2398:: — 1) dz :4. Jame-1 a: = [m u du =2Lu.n1:— r+G =lnlsucu+tanu1+ C 20. fish: 2+ cossz if: =Igflm¢=+m e'|+ C = [(241122 + 25in ECDS : + cos-2:) dz 25. [sainme dz: Heir: n {00911: in) = [Jr—i- Iuin 2: d: Billir+aalgr=1 u_ :2 andaniILsms-Eein‘zr '— zi-sinil? + C =::+%{-couf!=}+C 20. Izm’zdz= xtan s+1n1cos :I+ G : x — 13431523: -1- C Note: Here is an achrnabe {but equivalent] procedure. 21'. [tan‘i sec t d1: [fist-3:62! — 1:12:53: 1 I" IIEsin 3+ cos :32 dz: : [(m51_2m3r_ + m 1} d; = [(singr + 25in :ms z+wairj nix =flim+glaecat d! —2Iaec3t dt+ sec t I“ A = Id:+2]sinz{cmxdx} =§m3ttani-=}Jsec31di+lnlsec1+ Lam ti = r+ 2[1§(sin xfl+ C 1 5 5 3 :z+fl[ngx+0 = smitafli—Emitant+gllllsecl+mnfl+ C 21. [tannin sex: 1: in = [[scclu a Use: It du 25. [(21426)dfl =%lnlcsc 23 — ml; 291+ C = [[Mcau — sec 51}an Formula 423 29. [maroon dz: = - [mafia-(v—cacxcol, a: dc} =M—'T—“"L“"“+%Jsecuder-accudu =—51;nsc3::+6' 3!]. [max not“: dz: -%cot"’:+ C 31. Imixcusxdx= [w sin 3 = $36: :1 tan 1; — %II1[SEI: u + tan uI + C 22. Isa“: d: = [wearseczr d'a: Page 325 =-I[si|1:_1-l-C —m$+C 32. [tan mach dr: a: =J—EET cos rain .1: 2111 tan[:—r+§) 33. I 4 - Em: [1J4 F amiamma d9] = 41139523 [m :43 + sin 23] + C = 23in" g+§Mt — :‘+ C nix 3-1. _ iii; — a“ I l = sin“; + c By trigonometric substitution: let nt: 3 sin ti. Thtlt dz = 3 can {3 am, and V9~x2=V9—Qsinnfl=3mafl; I dz _ 3mg»th W" m : a + C=aiu‘1(§)+ G . 2+1 ul-E. :11 21rd: +J dz 2 iii-1+9 34+? =vm+lniat+ WHO By trigonometric substitution: lat .'I: = 2 tall 6'. Then dz = 2 secifl‘ :13, and 1H-l-;1=1H+4tnnl =29ec3; It“ nix: [2mmflglam29da] 7m :Zlmflmfldfii-Jsmfldfl =2mfl+lnlsec0+tanfll+61 =vm+infim+fl+q =m+1nim+xl+c as. [(Q-H‘P‘mdr Chapter .7, Methods of Intqgratiflu, =gnxg q-F+91n{:+ ¢9+¥Jl+c By trigonometric substitution: let a: = 3 tan 9. Then it: 3 sec-2i? W, and 139+?=V§+9tan!fl:3mfl; IJ9+FJI= Iamemmw d0] =915¢c3fldfl = W+1§1rflmcfi+mflfl+q =9[=V;[;i_; 1+§IHHVB + F +§|+ =§:¢W+gln( 9+ +=}—g1na IWHMHWIHG' aw. [aft—J =lnir+x/:“—-H+G By trigunnmutri: substitution: let a: = fl Then 11': = fist-c a tan a as. and W=W=fiuflfi d: _ fimfitfinfldfi' iE—?_iW =1mcfldfl=lnlsufl+tfln9|+cl :I x+'zfl_7 71" T =lnlg+1xz?_'rIu-§qu+cl =1nim+V12-7|+C d'J: ._ l l I aim-Hm)“ =ifitul1_l[‘/Ezj + 6 ti: : - —1 :7 c 39. 9m 76+ ll EL'.‘ + C: flu “'- imfi=iiflfq Section ?,3. Trigonomerrr'c Methods Page 32? =%m_1(—V;)+C =%ln|secl9+tan0l+0 41.[ if :y 3 + 1+1 +0 :9 4—1“ Jig—(awn? ;?9—(x+1]’ [ Econ-9dr] = . . =1 3+4 [dslngfl] -1 — 4&1n29 slhlm + C _ =%Iwc23dfl : 46. jv’ahfidfl Jim 4:“)? J1: I :-%cuLE+C= —___.V4"‘2+C 4: =JI'IJ'1 - sinifllfitw 3 d5} LEI. 3—1 '.= sin I? rJ'a: 42. f _ :3? + 9 = Imafldfl = $0 + 37511126 + r: =I 3mc23dfl “Jana 9t3n9+9 =aaiu_l(z—1}+%(:—1]V2£—xg+0 ,_ J until] dug _' ENE 41.- I J: =I «'r . 59-22% M; —1J=+5 =§Jcscdrifl 47—5—“5HW 5Lan€+5 = lmfldfi=lrflmfl+mnfll+vfll —§lnlcuc6'+mt 91+ C I n‘——-——-"'. tg+3l+c —2£+fi 3—] =1 T'FT'I‘C “l 5 5 1 'I‘lmiaamul:te'I||.'l-'LI.LII:I1lining";E — 2:+a+:-1|+ c d: 43- Ififi=[m =I'umflwnflfl IAt:—t-¢1=ufiscc£i 135ml] — 13 =Isecfldfi=1|flsecfl+tanfll+cl _ z+4+wJF+8s+3+C _ Gm 313 1 . a = I Bamazfldfl :Hdg ThismbewritmnaslnlmHHhC dlanfl+4 _l v—lr—l -3 _2m ( E )+C 49.1941115331:=Imauaccaudu=%tan4u+c man 45. 9—(r+1] “his: [mgr-i: 50. —%—= ——'—' Junx+wcs Elannz+l __ fiwnfld‘fl _1_ d6? _1 ‘Ig—usinisWvaVEIMEw Page 3'23 = ital—Ryan] + C = $751”: ' in}? tan 1*} + G 51. A = mixing: a: = ,1,(§]=% I} HIS 52. m = I we“: J: a: 0.9mm 53. r. L1 L1 56. 53. Eff-i NIB M = I Jams}: drflfi 0.0373155 r rrf-t 111.3 M,=%I m4: «1: 9:0.CIIESIJSQ wfd. (a?) as (-1.90, 0.19) ” r3 _ I 1 .— V_ er’nm 1: d:_ T 53 15-5031 0 a4. afit] = sing! um = Iain“: d: = y - gm: + c um]=%(fl] — isinfl+0=l€=2 w 2 5(1):“? — %sin2t+ 21d£=3f+21r 31835 0 IN] +cas :dx=[ 2magd: = Jfi‘coa gldx: i2flsin§r+ 0 [sin Brain 5: dz: Jamalt-rflt) — con-5:} d1: —:"-sin[ —-'2:::I —- fish} 32 + C = %sin 2: — fisin 31-F- C Ices sin 2: dz: = [1}[sin[gz}+ sing 3]] dz = —%ms[§x} - %cm[§=] + C Jena 7: mu( - flzlsin 44: d: = [fleas 10: + cos 4.1:]sin 42 :1: Chapter 3', Methods affair:ng ' = IiilsinE—fiz) + sin1421+§sin as} a: = [{Hflini - M + sin Ht] + {-ain 31'} fi- = filq’C‘JS'E-fifij - 515cm 14:: — 315mg: aficos 6:: — 3113:1019 Ha: — fines 33+ 0 59. Isinli‘l: cos 4:: If: : [£1 — cos Ethane 1 _ [Ewan - Ill-[cos ‘21: + :06 10:1} dz '- = gain 4: — 1gain '21: —n Tlfiain H]: + C “a. fll‘ = =}fr {gr- d': = I—atan 1'] d: In fl =%1I'IIDOB .1:I+ 01 f': 01.1“ch I’m} = 1 = 0.50:" = V‘coa : If: J‘f1+[1fws:h2dx u :12 :I 2:05:13“: 2 u 5 rd: 61' In—E— 9— '_.¢!t'.|1=\.llli--—z2;u2=£i—-z2 ' = -I d"1=_1rwg—?— 1|+c u— fh—u—du 1.! —H. 7.4 Hflhod of Partial Fractions, Pages-$55415? 1 I _-“‘I+ A2 ' :{3_ EJ—T 1._§ 1: Ad: 11 3} + Air Hr: u. then .41 :7}; if: = :3. than A, Thus. ...
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