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102-2006&amp;2007-2-M20-May2007

102-2006&amp;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) Iο¬ggddxβ 2. (4 points) Determine whether the following integral is convergent or divergent, and ο¬nd 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-,β 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"β. and use nit-egration by parts to get 1 5ββ (L )(11' = (1.1. 8111(eβ) + 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)] + Cβ / MM 2 8[β cos(a) + a, 3 β/4 β(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.β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 \$111β1(β:Lβ β 1) β sinβ1(- β 1) :: 3 _ sin'1(β?; _1 1 .J V317 β \$2 .143β 3 3 ββ¬βββββ 1β6") =limmΓ©oo β1 = β1. 3. (a) limzqoo 6ββ 111(1 β 6'3β) 2 limxaoo 8% 1_e_w βar:)eI _ l limzο¬ooο¬ β 6 ο¬ 8. (b) Appiying lβHospitalβ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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