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Unformatted text preview: Martinez, Gorge Exam 2 Due: Oct 31 2007, 1:00 am Inst: Diane Radin 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 1 3 x 9 x 2 x 2 dx. 1. I = 3 ln 2 2. I = 5 ln 3 3. I = 3 ln 3 4. I = 5 ln 2 5. I = 3 ln 3 6. I = 3 ln 2 7. I = 5 ln 2 correct 8. I = 5 ln 3 Explanation: After factorization x 2 x 2 = ( x + 1)( x 2) . But then by partial fractions, 3 x 9 x 2 x 2 = 4 x + 1 1 x 2 . Now Z 1 4 x + 1 dx = h 4 ln  ( x + 1)  i 1 = 4 ln 2 , while Z 1 1 x 2 dx = h ln  ( x 2)  i 1 = ln 2 . Consequently, I = 5 ln 2 . keywords: definite integral, rational function, partial fractions, natural log 002 (part 1 of 1) 10 points Evaluate the definite integral I = Z e 1 5 x 3 ln xdx. 1. I = 5 16 (3 e 4 + 1) correct 2. I = 15 16 e 4 3. I = 5 16 (3 e 4 1) 4. I = 5 4 (3 e 4 1) 5. I = 5 4 (3 e 4 + 1) Explanation: After integration by parts, I = h 5 4 x 4 ln x i e 1 5 4 Z e 1 x 3 dx = 5 4 e 4 5 4 Z e 1 x 3 dx, since ln e = 1 and ln 1 = 0. But Z e 1 x 3 dx = 1 4 ( e 4 1) . Consequently, I = 5 4 e 4 5 16 ( e 4 1) = 5 16 (3 e 4 + 1) . keywords: integration by parts, log function 003 (part 1 of 1) 10 points Martinez, Gorge Exam 2 Due: Oct 31 2007, 1:00 am Inst: Diane Radin 2 Evaluate the integral I = Z / 2 (1 + 2 sin 2 ) d . 1. I = correct 2. I = 1 4  1 3. I = 4. I = 1 2 5. I = 1 2  1 2 6. I = 1 1 4 Explanation: Since sin 2 = 1 2 1 cos 2 , the integral can be rewritten as I = Z / 2 n 2 cos 2 o d = h 2  1 2 sin 2 i / 2 . Consequently I = . keywords: definite integral, trig function, double angle formula 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z 2 t (3 t ) 2 dt. 1. I = 2 ln 3 correct 2. I = 3 ln 4 3. I = 2 + ln 3 4. I = 2(2 + ln 3) 5. I = 2(2 ln 3) 6. I = 2(3 ln 4) Explanation: Set u = 3 t . Then du = dt , while t = 0 = u = 3 , t = 2 = u = 1 . Then I = Z 1 3 (3 u ) u 2 du = Z 3 1 (3 u ) u 2 du = Z 3 1 n 3 u 2 1 u o du = h 3 u + ln  u  i 3 1 ....
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 Fall '09
 RAdin
 Calculus

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