HW06 Solution

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badruddin (ssb776) – HW06 – Radin – (57410) 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the integral I = integraldisplay e 1 1 x { 2 f (ln x ) + 3 } dx when f (0) = 1 and f (1) = 4. 1. I = 10 2. I = 11 3. I = 9 correct 4. I = 8 5. I = 12 Explanation: Set u = ln x . Then while x = 1 = u = 0 , x = e = u = 1 . In this case, I = integraldisplay 1 0 { 2 f ( u ) + 3 } du = integraldisplay 1 0 { 2 f ( u ) + 3 } du = bracketleftBig 2 f ( u ) + 3 u bracketrightBig 1 0 . Consequently, I = 2 { f (2) - f (1) } + 3 = 9 . 002 10.0 points Evaluate the definite integral I = integraldisplay 1 0 2 6 - 5 x dx . 1. I = 2 ln 6 5 2. I = 2 5 ln 11 6 3. I = 2 ln 11 6 4. I = 2 5 ln 6 correct 5. I = 2 ln 6 6. I = 2 5 ln 6 5 Explanation: Set u = 6 - 5 x ; then du = - 5 dx while x = 0 = u = 6 x = 1 = u = 1 . In this case, I = - 2 5 integraldisplay 1 6 1 du du = 2 5 integraldisplay 6 1 1 u du = 2 5 bracketleftBig ln | u | bracketrightBig 6 1 . Consequently, I = 2 5 (ln 6 - ln 1) = 2 5 ln 6 . 003 10.0 points Determine the indefinite integral I = integraldisplay x 3 x 2 + 3 dx . 1. I = x 2 + x ln( x 2 + 3) + C 2. I = 1 2 x 2 + 3 2 ln( x 2 + 3) + C 3. I = 1 2 x 2 - 3 2 ln( x 2 + 3) + C correct 4. I = 1 3 x 3 + 3 ln( x 2 + 3) + C

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badruddin (ssb776) – HW06 – Radin – (57410) 2 5. I = x 2 - x ln( x 2 + 3) + C 6. I = 1 3 x 3 - 3 ln( x 2 + 3) + C Explanation: After long division on the integrand, x 3 x 2 + 3 = x - 3 x x 2 + 3 . Thus I = integraldisplay x dx - integraldisplay 3 x x 2 + 3 dx . To evaluate the last integral we use the sub- stitution u = x 2 + 3. For then du = 2 x dx , so I = 1 2 x 2 - 3 2 integraldisplay 1 u du . Consequently, I = 1 2 x 2 - 3 2 ln( x 2 + 3) + C . 004 10.0 points Evaluate the definite integral I = integraldisplay 9 1 2 x ( x + 5) dx . 1. I = 2 ln 4 3 2. I = 2( 7 - 6) 3. I = 2(2 2 - 6) 4. I = 2 ln 7 6 5. I = 4 ln 4 3 correct 6. I = 4( 7 - 6) 7. I = 4(2 2 - 6) 8. I = 4 ln 7 6 Explanation: Set u 2 = x . Then 2 u du = dx , while x = 1 = u = 1 x = 9 = u = 3 .
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