Calculus Third Editon By Strauss, Bradley and Smith sec7.1

# Calculus (3rd Edition)

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Unformatted text preview: Fig: 31'! Chlpter 3", Method: of integratin- . 1 . 1 li'lntegratiun . 1.1 F n :_ a: 2 = J‘ﬂ_1dﬂ=llﬂﬂl+ C Nol::i}0linnl =‘—:F+3={=?—1] ".=']n{r‘ — 2:? +3] + C = 2t:z +-~5 It J a gﬂté‘ - 2:“ + 3}‘=[(4;3 - 4:] ﬂ'ﬂ =' H a" £1: —1-(:‘ — 29+3J“+G _~ .5':EE'[email protected]'-J‘_{:‘ + 4: + a] ‘ 1K2: + ‘11 d1] =="In|:’ + dx+3l + G -- d 1.1. E]: FI+=+1 J: 2 I + a: + Ill—llmrri- 1} dz] 4 rich-1.... . . w ﬁlﬂ(l2+l+1]+c [33% u = ﬂ _ __ _ 1H: _‘ ‘ p. In. 1—4:] a: E _ I It“ . . J“ _ zJ—ﬂllzt _ 1} dz] 9+9“? - I%:—'113(:I—:}‘1+G it _ [Manama-is . ‘3' 1373—?- a —'mq" =I—u4d'u gg-I'c a -, . =—§(1+mt=]5+c 14' [W 'W SICHM J’J, Review afSubstr'tuﬁan and Integration by Table :lzzﬂnx—%]+C é-m. [Insd'm - =z]n::-:I:+C =a'_1c"[:—r:_]‘_l+C 5' .[Hiﬁrsﬁ 25. = r1: — [Eaj-lllilr1+ bc“l+ c 7?; + Jim}? 1- %Inl:+ 9'? + lI-I- C [wit—m lilarmlllﬂ. + 1 4- ‘—'| 29. If 1 + + m 3"- : ‘T+ '3' 31. Je_ 4’sin 5: d: Fun-mu a. 492 32. : [16+25)_1e_4‘[—4ain 5: — 5cm 5:)+ C —4 sin 51— ﬁmaE-r 2*4-6 41:' [um-1m =I[,_1.=2 —§—ain‘1x+{zv‘1— 9+0 104-51)“th = b_llnll +llzl+ C 515. I: In x d: Formu a 502 ‘M. Page 313' [7—9312 a — =-VG2—zz+C [41mm (2 +135 _ (r+ 1)‘ =---5-- 4 +0 = ﬂaw 11"(u— 11+ 0' Inf—1+“: m =lgl{3£— EJW+E : c‘ ’d': ‘—u 415%: - 34-] + C : 1n 2:: dz Formula. 5112; u = 2 ¥In £5) ={(;)(2=]1(Inl2=|— 3+ 0 i'gvln 2: — g + C '1’ +6 :3 — {flail-F- =:'[+C' [mam = afln 41:}3 — 3 (In x}? d: F = :(In :33 — [rfh'l 1:] — 2zln z+2151 + c = xlnax — Eztnlt-f-ﬁzln .1: — 63-!- C [13“ Jézi+1 ﬁju_u2 in. 1. Few + 11'” + c l—II—I [I '—'—i —J *F Fag: 314 Chapter 3', Methods of Integration :14. J—E—M d1: Formula WEI; u = 2: Iuinsx ms‘r ch: = Iainiz: c0343: {sin 1: dz] 1f? _ ‘2 q — =\!¢1?.+1—In1+ 4 +1+C -—[[1-Cmr}muzl:3Inmdz.-j ix = Jms‘x (sin 2: dz] — Icon“: {sin 1: dz} 35. Inc-2.1%) dz Formula 42? 5 'i" : —Iu4du+JnEdu= —£5-+-¥‘,-+G _ g z :r z -sec2tan§+1nlacc§+tanﬁl+ﬂ =_%cmar+%wu71+c 35. Jain”: dz {4. Iain”: tong: d1: = [1— am; 2t][1+cns ‘23:) d: = —EBiI15IC03 J: 1- gfasin‘: 11': = — errant] d: : i—Iainih: dz __lrﬁ . 53_sin2: sin-1r _11_1- ._ 33m aromz+g[r 4 +—M—]+C —_a{§: gsm-ias)+0 I'ormuaﬁﬁl =§ W ﬁsh 413+ C = —&sinsx mu z+ ﬁr—ﬁsin2r+ “fisin 43+ C j d 415. lfrrlisodd.1ct u = cos a If n is odd, let 3? [Tn—“4' = mm 1.; = sin 1:. “both an and r1 are men, use the 9 + E: + 1 [3: + I) identities shown in Problems 41] and 4] until _ _ 1 + 0 one exponan is odd. '- 3E3r+1 :- l 46. Let u = c‘ﬂ; du = if” da: 33. [(g—ﬂmdz 1“ 3 2 = IK'JHIEJI +2m9—ﬁ1’2 =2J'I%=E[u-IHI1+HI+C 4 E +%§sin_ + C :25": — 2!n[1+erﬂ}+0 ginia: a I I I r 3!]. Iuﬁﬁ 15: _urmu|a .182 or use |_enLItI-25 4T. Lat u : :1“; d“ = %rnaﬂd: Z + a: = - ' 1 t. I E c — smx+ “I a“(§+4) + = [4.9 “HIE—«‘11; Formula“ H + 4|}- Iainaz d2: : d: 2 u+u 2 . : (“7+1_ 2(uJ-4-l]+lnlu+l|]+cl =%3 _ ésin2z+ C =4['§u2+u+% — 2n-2+lnlu+l|]+ cl =%3‘ — %3E11 2:+ C 41. JcmadnJl—Lciyﬁd, =4Eu2—u+lnlu+1[l-I—C ' 12 =vls= +%B-—g—"12:+ C = 52;— — It“ +1I1(::”" +1]]+ C 1 I' : —.r:+-sm 23+ C «1d: _ 6. _ a ' 2 ‘ 43. I—F—wgrlz Iatr—u,dﬁ:_ﬁu til: i 42. Lctu = sins, that = matrix“, 5 _ I 4mm] _ 24 m i [sin‘xcoardr = %—+0 = isin5x+ﬂ _ 1;"! + 24:3 _ u2 + 2a: '13. Lat 11 = cos :. du = —ain 1dr. '_' 2" lg" d1; _ W4“ W2 muld use Formula 44 from the Sludcm! Section 12!. Review of Subslr'tutl'an and Integration by Table Page 3J5 gls} _ «ng(%) Mathematics Handbook (or other table of = x 0 _ﬂ_— integrals], but ainec the degree of the 9 numeth is greater than the degree of the _ 3| I _ sﬂﬂ “1 9c. denominator, do long division to obtain: _ T E a I 81 9 lﬂrl 1__ E =-:1-1r-§'K{ln9[l—]nﬂ] MIG“ 4“ + ﬁ inf—1)“ 81 _ n3 u: n ] = T! — 91? In 1|] F: GILES- —2v1(f—-3—+E—mln|2u+ll)-l-C i __ 132; f3 Jana lfﬁ ‘ _ 4:. Jr] + 3:1 Int}: + l] + C at V: “Illa ; Eff-1.12 dx 1% Lana! Mezt u '49. j—-—r d: (2 + Lanai) I Let. u = 2 + tan :3. tanzt 396:2! d: = wJ'EU _ 3231!? «I You could use Formuta. 152 from the Student Mathematics Handinok (or other table of integrals}; inntead. let :r = sin f}, it = coal? Jr? and 1— §=coaﬂ;i[z=0,thenﬂ=ﬂ andifx: 1,3: 9’2: 113'! v: «I sinia on: 90;an an) El “N 2 amines) = 11']- 4 Jill 4 _. l 1,:‘3 55. V—21!1{7;(1+ VG) ]d: 1 = 21J’ﬁ(l+ vial-’3 a: J = gases «ii/5 — as W} 9: 40.3053 Use it 2 1+ J; or amlculstor 2 2 55. v: rrIHVﬁ— F)“ dy: «In # 923.1” dy 1 1 Forrnu a. 231 _ 21;: 2 = ﬂ‘g—‘iy—J—‘FgﬂjD-l%] 1 =05 Page 315 = aha-1x — 31,15} :3 3.3590 51'. ty' = "‘[In t]: — a? (In)? —x‘ _ __?__ 1;: 2 1:2 5:] 1+v“i:;l-l=—J.!ﬂf!dt Iji If! u =1n 1:; d1: = Q? J: or acalcuintor = Fyﬁﬁ = gluzﬂ :3 0.1207 1!! 58. dy: -EL‘;:de= --tanzdr, da=Vl+tanlzdz.-_-sec:de 1r,“ 5: Jamxdx=lnleeez+tau IHKS‘ IJ =ln(l + «ﬂy a: 0.8814 59. d3: \g‘l +4? chi; dS=2ryds I 2 2 s: zxjéy’l-H? d1: = 2:]“Tv’1+u*(%) I] I} Formula 17': 1.: r: 2: _ If“? +1331” m uh? +1314” 4 d E 3 4%! u+[u"+1‘ll"2]) u 9J5, :Intﬁm) 15 ' T2— Or, by calculator S as 3.8!]91' GD. nip—:2: dz; ds=1,-"1+«i::I dz; d3=2nuda 1 ﬂew—7H4 as {J = 3%)(1 + 4333321; = \$151.13 — 1} Or, by calculator. 3 =2 5.33011 Chapter I. Methods of {Meg-reﬁne —cuc mfmz+cot ﬁll 61. Imed':=-—[ mx+mt= = —1n]eacx+cotrl+ C 62. a. I25in£mstdr= —2J'uos.v(-sin: = —2{1ﬁ)coa2:+ CI = emszz+ OK I b. J-‘lainrcoa :dz:2[aiu :{coexds} = 2(%}ain2:+ c2 = sin“: + c: c. Jﬂainxmrdr=Iaiuizdx = —}§cou2£+ C: d. All three are the same with C1 = Cf!- 62:03 _ IE1 =Ga+%u 63. Letu=sr—::,d'u:—d'1:. Thcnsin== 11' D Isﬂain :] dz=J.(1r — ujﬂain u]{-— Eu) 3' D [-11- - u) ffsin «Nu 1r Kain u} du — [1: Hail: 1:] du 13 The value of the integral is independent of variable used 30 I I = «lﬂsﬁn .13} d3: — lzﬂuin x) 6.1: Add the second integral Lo both sides and divide by 2 to obtain the desired result. E4. The surface area S or the torus is the sum area 5'1 generated I tating the top of circley= i: + V1 — 915 about them the area 3:, generated by rotating the be D"—'|=l CF—IH y=b— 1— z’aboutLhcsameaxia. either case, dy lxl ‘.'I 1 :1 = “(35 +(Ul-E) 1+? Wehm'm S: 31+S; =12!” + FEM/1:37? a: Section F2. Integration by Part: Jr]. 2:“ — m; XﬁL-Fdr 1 “I _ 41b . —] 1 _ d1: = arbsm II J 1 — :2 #1 ...
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