M408L hwk6 solutions

# M408L hwk6 solutions - Pham, Quoc Homework 6 Due: Feb 27...

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Unformatted text preview: Pham, Quoc Homework 6 Due: Feb 27 2007, 3:00 am Inst: Eric Katerman 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Evaluate the definite integral I = Z / 4 3 cos x- 4 sin x cos 3 x dx. 1. I = 1 2 2. I = 0 3. I = 1 correct 4. I = 3 2 5. I =- 1 2 Explanation: After division 3 cos x- 4 sin x cos 3 x = 3 sec 2 x- 4 tan x sec 2 x = (3- 4 tan x ) sec 2 x. Thus I = Z / 4 (3- 4 tan x ) sec 2 xdx. Let u = tan x ; then du = sec 2 xdx so I = Z 1 (3- 4 u ) du = 3 u- 2 u 2 / 1 . Consequently, I = 1 . keywords: substitution, trigonometric identi- ties 002 (part 1 of 1) 10 points Evaluate the indefinite integral I = Z 2cos 4 3 tdt. 1. I = 1 4 3 t + 2 3 cos 6 t + 1 12 cos 12 t + C 2. I = 1 4 3 t- 2 3 sin6 t + 1 12 sin 12 t + C 3. I = 1 4 3 t + 2 3 cos 6 t- 1 12 cos 12 t + C 4. I = 1 4 3 t + 2 3 sin 6 t + 1 12 sin12 t + C correct 5. I = 1 4 3 t- 2 3 cos 6 t + 1 12 cos 12 t + C 6. I = 1 4 3 t + 2 3 sin6 t- 1 12 sin 12 t + C Explanation: Since cos 2 = 1 2 1 + cos 2 , the integrand can be rewritten as 2cos 4 3 t = 1 2 1 + cos 6 t 2 = 1 2 1 + 2cos 6 t + cos 2 6 t . But in turn, this last expression can be rewrit- ten as 1 2 1 + 2cos 6 t + 1 2 n 1 + cos 12 t o . Thus 2cos 4 3 t = 1 2 3 2 + 2cos 6 t + 1 2 cos 12 t , Pham, Quoc Homework 6 Due: Feb 27 2007, 3:00 am Inst: Eric Katerman 2 and so I = 1 2 Z 3 2 + 2cos 6 t + 1 2 cos 12 t dt. Consequently, I = 1 4 3 t + 2 3 sin6 t + 1 12 sin 12 t + C with C an arbitrary constant. keywords: linearization of trigonometric func- tions 003 (part 1 of 1) 10 points Evaluate the definite integral I = Z / 4 sin2 ( 1- sin 2 ) d . 1. I = 7 8 2. I = 5 8 3. I = 3 8 correct 4. I = 1 2 5. I = 1 8 Explanation: Since sin 2 = 1 2 (1- cos 2 ) , we see that 1- sin 2 = 1- 1 2 (1- cos 2 ) = 1 2 + 1 2 cos 2 Thus I = 1 2 Z / 4 sin2 (1 + cos 2 ) d ....
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## This homework help was uploaded on 03/19/2008 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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M408L hwk6 solutions - Pham, Quoc Homework 6 Due: Feb 27...

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