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102-2006&2007-2-M20-May2007

102-2006&2007-2-M20-May2007 - Kuwait University...

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Unformatted text preview: Kuwait University Math 102 Date: 03/05/2007 Dept. ofMath 8; Computer. Sci. {Seward F-Fidterrn} Fβ€˜vr::t'Β§:r.=.tr 90 it'2ir':iβ€˜:f.;*s (Answer all the following questions. Calculators and mobile phones are not allowed) I. (3 points each) Evaluate the following integrals a) Iflggddxβ€˜ 2. (4 points) Determine whether the following integral is convergent or divergent, and find its value in the case of convergence. 3 i I __;_ (if. 1 3x β€” x2 3. (3 points each) a) Find the following limit if it exists " lim(lβ€˜β€” ed)" 1-,” b) Find b such that . sin 3x β€” 3x + bx3 -. 11m 3 = 0 xβ€”m x (hunt) (a) / Maia: 8'37 I n ' β€˜. "I ryr ,. n ,w Take t = 6"”. and use nit-egration by parts to get 1 5β€˜β€ (L )(11' = (1.1. 8111(eβ€œ) + V 1 _ FZT + C. ' sec .17 (b) / ( )441- . tan .T _ _ j) 7_ ') _ β€˜7 sec 1: . 4 seeβ€˜ 1: secβ€” 1? 1 + u- _ . ( ) d1: = β€”β€”β€”4β€”β€”d.1' = 4 (With it : seer and nsmg tan 1: tan .1: u 1 + tan2 :v 2 sec2 17.), So, 1: seer 4 β€”1 _1 .i d;rβ€” β€” β€”β€” tan a? β€” tan :1: C. [(tanat) 3 + (C)/~β€”-β€”-β€”β€”β€”- (In x)3 23β€˜ /4 β€” (1n 11:)2dx Take t = 1n :6, and use trigonometric substitution t = sin(a), to obtain 1(305302)] + C’ / MM 2 8[β€” cos(a) + a, 3 β€˜/4 β€œ(11120)? 3132+1β€˜ (d)/ x4_1dx The decomposition in partial fractions gives 3x2+1_ 1 1 _ 1 234-1 113β€”1 x+1+x2+1 Hence, (in: = ln(a: β€” 1) β€” ln(:c + 1) + tamβ€”I a: + C. /3a:2+1 234β€”1 CO / 00th a: / 0051β€˜ :3 d3: = secβ€”1(sinh 9:) + C. β€”β€”-β€”-β€”β€”β€”-β€”d.’Lβ€˜ = Vcoshz x β€” 2 sinh 1123/ sinh2 a: β€” 1 2. 3m β€” x2 = g β€” (a: β€” %)2 Take a: β€” g = gsinm) to obtain 3 d 2 2 u 2 / ._β€”β€”β€”$β€”β€” : 11m $111”1(β€”:Lβ€˜ β€” 1) β€” sinβ€”1(- β€” 1) :: 3 _ sin'1(β€”?; _1 1 .J V317 β€” $2 .143” 3 3 β€”β‚¬β€”β€”β€”β€˜β€œ 1β€˜6") =limmΓ©oo β€˜1 = β€”1. 3. (a) limzqoo 6”” 111(1 β€” 6'3”) 2 limxaoo 8% 1_e_w β€”ar:)eI _ l limzfloofl β€” 6 fl 8. (b) Appiying l’Hospital’s rule three times, we obtain lim sin(3x) β€” 3x + bx3 = β€”-27 + 6b = 0 wβ€”vo :13? 6 So, b [ONO II 00 CO ...
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