HW 8 Solution - Kim Jin – Homework 8 – Due 1:00 pm –...

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Unformatted text preview: Kim, Jin – Homework 8 – Due: Oct 24 2007, 1:00 pm – Inst: Diane Radin 1 This print-out should have 24 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 1- 1 e 6 arctan y 1 + y 2 dy . 1. I =- 1 7 e 3 π/ 2 + 1 7 e- 3 π/ 2 2. I = 1 6 e 3 π/ 2 + 1 6 e- 3 π/ 2 3. I = 1 7 e 3 π/ 2 + 1 7 e- 3 π/ 2 4. I = 1 6 e 3 π/ 2- 1 6 e- 3 π/ 2 correct 5. I =- 1 6 e 3 π/ 2- 1 6 e- 3 π/ 2 6. I = 1 7 e 3 π/ 2- 1 7 e- 3 π/ 2 Explanation: Set u = arctan y . Then du = 1 1 + y 2 dy , in which case I = Z π/ 4- π/ 4 e 6 u du = h e 6 u 6 i π/ 4- π/ 4 . Consequently, I = 1 6 ( e 3 π/ 2- e- 3 π/ 2 ) . keywords: substitution, inverse trig function, integral 002 (part 1 of 1) 10 points Evaluate the definite integral I = Z 2 1 x 2 + 4 x + 1 dx. Correct answer: 2 . 52733 . Explanation: After division x 2 + 4 x + 1 = ( x 2- 1) + 5 x + 1 = x 2- 1 x + 1 + 5 x + 1 = x- 1 + 5 x + 1 . In this case I = Z 2 1 ‡ x- 1 + 5 x + 1 · dx = h 1 2 x 2- x + 5 ln | x + 1 | i 2 1 = ‡ 1- 1 2 · + 5 ‡ ln3- ln2 · . Consequently, I = 1 2 + 5 ln 3 2 = 2 . 52733 . keywords: division, log, integral 003 (part 1 of 1) 10 points Evaluate the integral I = Z π/ 4 sec 2 x { 2- 3 sin x } dx. 1. I =- 1- 3 2 √ 2 2. I = 5 + 3 √ 2 3. I = 5 + 3 2 √ 2 4. I =- 1 + 3 2 √ 2 5. I = 5- 3 √ 2 correct Kim, Jin – Homework 8 – Due: Oct 24 2007, 1:00 pm – Inst: Diane Radin 2 6. I =- 1- 3 √ 2 Explanation: Since sec 2 x { 2- 3 sin x } = 2 sec 2 x- 3 sec x ‡ sin x cos x · , we see that I = Z π/ 4 { 2 sec 2 x- 3 sec tan x } dx. But d dx tan x = sec 2 x, while d dx sec x = sec x tan x. Consequently, I = h 2 tan x- 3 sec x i π/ 4 = 5- 3 √ 2 . keywords: definite integral, tan integral, sec integral 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z 1 √ 2 x 2 √ 1- x 2 dx. 1. I = π 2- 1 4 2. I = π- 1 2 3. I = π 4- 1 8 4. I = π 2- 1 5. I = π 4- 1 2 6. I = π 8- 1 4 correct Explanation: Set x = sin u . Then dx = cos udu, p 1- x 2 = cos u, while x = 0 = ⇒ u = 0 , x = 1 √ 2 = ⇒ u = π 4 . In this case I = Z π/ 4 sin 2 u cos u cos u du = Z π/ 4 sin 2 udu = 1 2 Z π/ 4 ‡ 1- cos 2 u · du. Thus I = 1 2 h u- 1 2 sin 2 u i π/ 4 . Consequently, I = 1 8 π- 1 4 . keywords: definite integral, trig. substitution, half-angle identity 005 (part 1 of 1) 10 points Stewart Chap. 8, sect. 5, Ex 5 page 545 Evaluate the definite integral I = Z √ 3 r 2- x 2 + x dx. 1. I = 2 3 π- √ 3 2. I = 1 3 π + √ 3 Kim, Jin – Homework 8 – Due: Oct 24 2007, 1:00 pm – Inst: Diane Radin 3 3. I = 2 3 π + 1 4. I = 1 3 π- √ 3 5. I = 2 3 π- 1 correct 6. I = 1 3 π + 1 Explanation: Rationalizing the numerator we see that r 2- x 2 + x = 2- x √ 2 + x √ 2- x = 2- x √ 4- x 2 ....
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This note was uploaded on 04/12/2010 for the course PHY 58195 taught by Professor Turner during the Spring '09 term at University of Texas.

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HW 8 Solution - Kim Jin – Homework 8 – Due 1:00 pm –...

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