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Exam2Sol

# Exam2Sol - Massaro Michael Exam 2 Due 1:00 am Inst Shinko...

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Massaro, Michael – Exam 2 – Due: Oct 31 2007, 1:00 am – Inst: Shinko Harper 1 This print-out should have 16 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 0 5 x - 7 x 2 - x - 2 dx . 1. I = 5 ln 3 2. I = 3 ln 2 correct 3. I = - 5 ln 2 4. I = - 3 ln 2 5. I = - 5 ln 3 6. I = - 3 ln 3 7. I = 5 ln 2 8. I = 3 ln 3 Explanation: After factorization x 2 - x - 2 = ( x + 1)( x - 2) . But then by partial fractions, 5 x - 7 x 2 - x - 2 = 4 x + 1 + 1 x - 2 . Now Z 1 0 4 x + 1 dx = h 4 ln | ( x + 1) | i 1 0 = 4 ln 2 , while Z 1 0 1 x - 2 dx = h ln | ( x - 2) | i 1 0 = - ln 2 . Consequently, I = 3 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 2 x 2 ln x dx. 1. I = 2 3 (2 e 3 - 1) 2. I = 2 3 (2 e 3 + 1) 3. I = 2 9 (2 e 3 + 1) correct 4. I = 4 9 e 3 5. I = 2 9 (2 e 3 - 1) Explanation: After integration by parts, I = h 2 3 x 3 ln x i e 1 - 2 3 Z e 1 x 2 dx = 2 3 e 3 - 2 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 = 2 3 e 3 - 2 9 ( e 3 - 1) = 2 9 (2 e 3 + 1) . keywords: integration by parts, log function 003 (part 1 of 1) 10 points

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Massaro, Michael – Exam 2 – Due: Oct 31 2007, 1:00 am – Inst: Shinko Harper 2 Evaluate the integral I = Z π/ 4 0 (1 + 2 sin 2 θ ) dθ . 1. I = 1 - 1 4 π 2. I = π 3. I = 1 4 π - 1 4. I = - π 5. I = - 1 2 π 6. I = 1 2 π - 1 2 correct Explanation: Since sin 2 θ = 1 2 1 - cos 2 θ · , the integral can be rewritten as I = Z π/ 4 0 n 2 - cos 2 θ o = h 2 θ - 1 2 sin 2 θ i π/ 4 0 . Consequently I = 1 2 π - 1 2 . keywords: definite integral, trig function, double angle formula 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z 2 0 t (3 - t ) 2 dt . 1. I = 3 - ln 4 2. I = 2 + ln 3 3. I = 2(3 - ln 4) 4. I = 2(2 - ln 3) 5. I = 2 - ln 3 correct 6. I = 2(2 + ln 3) 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 . Consequently, I = - (1 - 3) - ln 3 = 2 - ln 3 . keywords: substitution, integral 005 (part 1 of 1) 10 points To which one of the following does the inte- gral I = Z x x 2 - 1 dx reduce after an appropriate trig substitution.
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Exam2Sol - Massaro Michael Exam 2 Due 1:00 am Inst Shinko...

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