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Unformatted text preview: Valenica, Daniel Exam 2 Due: Oct 31 2007, 1:00 am Inst: Cheng 1 This printout should have 16 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 3 3 x 7 x 2 3 x 4 dx. 1. I = 3 ln 5 2. I = 3 ln 5 3. I = ln 5 4. I = ln 5 5. I = ln 4 correct 6. I = ln 4 7. I = 3 ln 4 8. I = 3 ln 4 Explanation: After factorization x 2 3 x 4 = ( x + 1)( x 4) . But then by partial fractions, 3 x 7 x 2 3 x 4 = 2 x + 1 + 1 x 4 . Now Z 3 2 x + 1 dx = h 2 ln  ( x + 1)  i 3 = 2 ln 4 , while Z 3 1 x 4 dx = h ln  ( x 4)  i 3 = ln 4 . Consequently, I = ln 4 . keywords: definite integral, rational function, partial fractions, natural log 002 (part 1 of 1) 10 points Evaluate the definite integral I = Z e 1 3 x 2 ln xdx. 1. I = (2 e 3 1) 2. I = (2 e 3 + 1) 3. I = 1 3 (2 e 3 + 1) correct 4. I = 1 3 (2 e 3 1) 5. I = 2 3 e 3 Explanation: After integration by parts, I = h x 3 ln x i e 1 Z e 1 x 2 dx = e 3 Z e 1 x 2 dx, since ln e = 1 and ln 1 = 0. But Z e 1 x 2 dx = 1 3 ( e 3 1) . Consequently, I = e 3 1 3 ( e 3 1) = 1 3 (2 e 3 + 1) . keywords: integration by parts, log function 003 (part 1 of 1) 10 points Valenica, Daniel Exam 2 Due: Oct 31 2007, 1:00 am Inst: Cheng 2 Evaluate the integral I = Z / 4 (1 4 sin 2 ) d . 1. I = 1 4  1 2. I = 1 2  1 2 3. I = 1 2 4. I = 5. I = 1 1 4 correct 6. I = Explanation: Since sin 2 = 1 2 1 cos 2 , the integral can be rewritten as I = Z / 4 n 2 cos 2  1 o d = h sin 2  i / 4 . Consequently I = 1 1 4 . keywords: definite integral, trig function, double angle formula 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z 1 t (2 t ) 2 dt. 1. I = 2(2 ln 3) 2. I = 1 + ln 2 3. I = 2 ln 3 4. I = 2(1 + ln 2) 5. I = 1 ln 2 correct 6. I = 2(1 ln 2) Explanation: Set u = 2 t . Then du = dt , while t = 0 = u = 2 , t = 1 = u = 1 . Then I = Z 1 2 (2 u ) u 2 du = Z 2 1 (2 u ) u 2 du = Z 2 1 n 2 u 2 1 u o du = h 2 u + ln  u  i 2 1 ....
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 Fall '07
 CHENG

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