AnsHW7 - Cheung, Anthony Homework 7 Due: Oct 17 2006, 3:00...

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Unformatted text preview: Cheung, Anthony Homework 7 Due: Oct 17 2006, 3:00 am Inst: David Benzvi 1 This print-out should have 17 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 3 6 x 2 + 1 dx. 1. I = 3(2- 3 ) 2. I = 6ln(2- 3) 3. I = 6ln(2 + 3) correct 4. I = 6(2- 3 ) 5. I = 3 ln(2 + 3) 6. I = 6(2 + 3 ) Explanation: Set x = tan u, then dx = sec 2 udu, x 2 + 1 = sec 2 u, while x = 0 = u = 0 , x = 3 = u = 3 . In this case I = Z 3 6sec 2 u sec u du = Z 3 6sec udu. Now Z sec udu = ln | sec u + tan u | + C 1 , Thus I = 6 h ln | sec u + tan u | i / 3 . Consequently, I = 6ln(2 + 3 ) . keywords: 002 (part 1 of 1) 10 points Evaluate the integral I = Z 1 2 t 2- t 4 dt. 1. I = 2. I = 1 3. I = 1 2 4. I = 1 4 5. I = 1 4 correct 6. I = 1 2 Explanation: Set t 2 = 2 sin u . Then 2 tdt = 2cos udu, in which case tdt = 1 2 dt. On the other hand, t = 0 = u = 0 , t = 1 = u = 4 . Cheung, Anthony Homework 7 Due: Oct 17 2006, 3:00 am Inst: David Benzvi 2 Thus I = 1 2 Z / 4 2cos u 2cos u du = 1 2 h 2 u i / 4 . Consequently, I = 1 4 . keywords: definite integral, substitution, in- verse sin integral, 003 (part 1 of 1) 10 points Evaluate the definite integral I = Z 4 2 7 x 2- 6 x + 10 dx. 1. I = 4 2. I = 7 2 correct 3. I = 7 4. I = 3 5. I = 3 2 Explanation: By completing the square we see that x 2- 6 x + 10 = ( x 2- 6 x + 9) + 10- 9 = ( x- 3) 2 + 1 . Thus I = Z 4 2 7 ( x- 3) 2 + 1 . Since d dx tan- 1 x = 1 1 + x 2 , this suggests the substitution x- 3 = u . For then dx = du , while x = 2 = u =- 1 , x = 4 = u = 1 . In this case, I = 7 Z 1- 1 1 1 + u 2 du = 7 h tan- 1 u i 1- 1 = 7 tan- 1 1- tan- 1 (- 1) / . But tan- 1 1 = 4 while tan- 1 (- 1) =- 4 . Consequently, I = 7 2 . keywords: trigonometric substitutions, com- plete square 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z 1 / 2 2 sin- 1 x + 5 1- x 2 dx. 1. I = 2 + 2 + 3 2. I = 2 - 2 + 3 3. I = 2 + 2- 3 4. I = - 2- 3 correct 5. I = + 2- 3 Explanation: Let x = sin ; then dx = cos d and 1- x 2 = 1- sin 2 = cos 2 , while x = 0 = = 0 , x = 1 2 = = 6 . Cheung, Anthony Homework 7 Due: Oct 17 2006, 3:00 am Inst: David Benzvi 3 In this case I = Z / 6 2 + 5 cos cos d = Z / 6 (2 cos + 5) d = 2 Z / 6 cos d + 5 6 ....
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AnsHW7 - Cheung, Anthony Homework 7 Due: Oct 17 2006, 3:00...

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