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Unformatted text preview: Portillo, Tom Homework 6 Due: Oct 10 2006, 3:00 am Inst: David Benzvi 1 This printout should have 17 questions. Multiplechoice 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 (6 x + 5) sec 2 xdx. 1. I = 3 2  5 3 ln 2 2. I = 3  5 + 3 ln 2 3. I = 3  5 + 6 ln 2 4. I = 3 2 + 5 3 ln 2 correct 5. I = 3 + 5 6 ln 2 Explanation: Since d dx tan x = sec 2 x, integration by parts is suggested. For then, I = h (6 x + 5) tan x i / 4 Z / 4 tan x d dx (6 x + 5) dx. = 3 2 + 5 6 Z / 4 tan xdx. But Z / 4 tan xdx = h ln  sec x  i / 4 = ln 2 , so I = 3 2 + 5 3 ln 2 . keywords: integration by parts, trig function 002 (part 1 of 1) 10 points Determine the indefinite integral I = Z e x cos xdx. 1. I = 1 2 e x sin x + cos x + C correct 2. I = e x sin x cos x + C 3. I = 1 2 e x sin x cos x + C 4. I = e x sin x + cos x + C 5. I = e x sin x cos x + C 6. I = 1 2 e x sin x + cos x + C Explanation: After integration by parts, I = e x cos x Z e x d dx cos xdx = e x cos x + Z e x sin xdx. To reduce this last integral to one having the same form as I , we integrate by parts again for then Z e x sin xdx = e x sin x Z e x d dx sin xdx = e x sin x Z e x cos xdx = e x sin x I . Thus I = e x cos x + n e x sin x I o . Solving for I we see that 1 + 1 I = e x cos x + e x sin x. Portillo, Tom Homework 6 Due: Oct 10 2006, 3:00 am Inst: David Benzvi 2 Consequently I = 1 2 e x sin x + cos x + C with C an arbitrary constant. keywords: indefinite integral, integration by parts, exponential function, cosine function 003 (part 1 of 1) 10 points Determine the integral I = Z 5 ln x x 4 dx. 1. I = 5 3 x 3 ln x + 1 3 + C correct 2. I = 5 4 x 3 ln x + 1 3 + C 3. I = 5 3 x 3 ln x + 1 3 + C 4. I = 5 3 x 3 ln x 1 3 + C 5. I = 5 4 x 3 ln x 1 3 + C 6. I = 5 4 x 3 ln x + 1 3 + C Explanation: After integration by parts Z ln x x 4 dx = 1 3  ln x x 3 + Z 1 x 4 dx = 1 3 x 3 ln x + 1 3 + C . Consequently, I = 5 3 x 3 ln x + 1 3 + C with C an arbitrary constant. keywords: indefinite integral, log integral, in tegration by parts 004 (part 1 of 1) 10 points Determine the indefinite integral I = Z 3 cos(ln x ) dx....
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This note was uploaded on 02/18/2010 for the course MATH 408L taught by Professor Benzvi during the Spring '06 term at North Texas.
 Spring '06
 Benzvi

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