# HW_6_Answers - Miller Kierste – Homework 6 – Due 3:00...

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Unformatted text preview: Miller, Kierste – Homework 6 – Due: Feb 27 2007, 3:00 am – Inst: Gary Berg 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 π/ 4 cos x- 3 sin x cos 3 x dx. 1. I = 1 2. I = 3 2 3. I = 0 4. I = 1 2 5. I =- 1 2 correct Explanation: After division cos x- 3 sin x cos 3 x = sec 2 x- 3 tan x sec 2 x = (1- 3 tan x ) sec 2 x. Thus I = Z π/ 4 (1- 3 tan x ) sec 2 xdx. Let u = tan x ; then du = sec 2 xdx so I = Z 1 (1- 3 u ) du = • u- 3 2 u 2 ‚ 1 . Consequently, I =- 1 2 . keywords: substitution, trigonometric identi- ties 002 (part 1 of 1) 10 points Evaluate the indefinite integral I = Z 2 cos 4 2 tdt. 1. I = 1 4 ‡ 3 t- cos 4 t + 1 8 cos 8 t · + C 2. I = 1 4 ‡ 3 t +sin 4 t + 1 8 sin 8 t · + C correct 3. I = 1 4 ‡ 3 t- sin 4 t + 1 8 sin 8 t · + C 4. I = 1 4 ‡ 3 t + cos 4 t + 1 8 cos 8 t · + C 5. I = 1 4 ‡ 3 t + cos 4 t- 1 8 cos 8 t · + C 6. I = 1 4 ‡ 3 t + sin 4 t- 1 8 sin 8 t · + C Explanation: Since cos 2 θ = 1 2 ‡ 1 + cos 2 θ · , the integrand can be rewritten as 2 cos 4 2 t = 1 2 ‡ 1 + cos 4 t · 2 = 1 2 ‡ 1 + 2 cos 4 t + cos 2 4 t · . But in turn, this last expression can be rewrit- ten as 1 2 ‡ 1 + 2 cos 4 t + 1 2 n 1 + cos 8 t o· . Thus 2 cos 4 2 t = 1 2 ‡ 3 2 + 2 cos 4 t + 1 2 cos 8 t · , Miller, Kierste – Homework 6 – Due: Feb 27 2007, 3:00 am – Inst: Gary Berg 2 and so I = 1 2 Z ‡ 3 2 + 2 cos 4 t + 1 2 cos 8 t · dt. Consequently, I = 1 4 ‡ 3 t + sin 4 t + 1 8 sin 8 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 π/ 4 sin 2 θ ( 4 + 2 cos 2 θ ) dθ . 1. I = 9 4 2. I = 5 4 3. I = 2 4. I = 7 4 5. I = 11 4 correct Explanation: Since cos 2 θ = 1 2 (1 + cos 2 θ ) , we see that 4 + 2 cos 2 θ = 4 + (1 + cos 2 θ ) = 5 + cos 2 θ Thus I = 1 2 Z π/ 4 sin 2 θ (10 + 2 cos 2 θ ) dθ ....
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HW_6_Answers - Miller Kierste – Homework 6 – Due 3:00...

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