HW 08 solution - gutierrez (ig3472) HW08 Neitzke (56585) 1...

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Unformatted text preview: gutierrez (ig3472) HW08 Neitzke (56585) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find the value of the integral I = integraldisplay 3 1 1 4 + ( x- 1) 2 dx . 1. I = 2 2. I = 1 2 3. I = 1 4 4. I = 2 5. I = 1 4 6. I = 1 8 correct Explanation: Set 2 tan u = x- 1. Then 4 + ( x- 1) 2 = 4 + (2 tan u ) 2 = 4(1 + tan 2 u ) = 4 sec 2 u , while 2 sec 2 u du = dx . Also x = 1 = u = 0 , and x = 3 = u = 4 . In this case I = integraldisplay / 4 2 sec 2 u 4 sec 2 u du = 1 2 integraldisplay / 4 du. Consequently, I = 1 2 bracketleftBig u bracketrightBig / 4 = 1 8 . 002 10.0 points Evaluate the integral I = integraldisplay 1 / 2 sin- 1 x 1- x 2 dx . 1. I = 2 72 correct 2. I = 2 4 3. I = 2 8 4. I = 2 9 5. I = 2 18 Explanation: Set x = sin u . Then dx = cos u du, 1- x 2 = cos 2 u , while x = 0 = u = 0 x = 1 2 = u = 6 . In this case I = integraldisplay 6 u cos u cos u du = integraldisplay 6 u du . Consequently, I = bracketleftbigg u 2 2 bracketrightbigg 6 = 2 72 . 003 10.0 points Determine the integral I = integraldisplay 1 ( x 2 + 4) 3 2 dx . 1. I = x 4 x 2 + 4 + C correct gutierrez (ig3472) HW08 Neitzke (56585) 2 2. I = 1 4 x 2 + 4 + C 3. I = x x 2 + 4 + C 4. I = x 2 + 4 4 x + C 5. I = x 2 + 4 x + C 6. I = x x 2 + 4 4 + C Explanation: Set x = 2 tan u. Then dx = 2 sec 2 u du , while ( x 2 + 4) 3 2 = ( 4(tan 2 u + 1) ) 3 2 = 8 sec 3 u . Thus I = integraldisplay 2 8 sec 2 u sec 3 u du = 1 4 integraldisplay cos u du , and so I = 1 4 sin u + C = 1 4 sin parenleftBig tan- 1 x 2 parenrightBig + C . But by Pythagoras u radicalbig x 2 + 4 2 x we see that sin parenleftBig tan- 1 x 2 parenrightBig = x x 2 + 4 . Consequently, I = x 4 x 2 + 4 + C with C an arbitrary constant. keywords: trig substitution 004 10.0 points Evaluate the definite integral I = integraldisplay 2 3 x 2 4 + x 2 dx . 1. I = 3 4 ( + 2) 2. I = 3 2 (4- ) correct 3. I = 3 4 ( - 2) 4. I = 3 2 ( - 2) 5. I = 3 2 (4 + ) 6. I = 3 4 (4- ) Explanation: Let x = 2 tan ; then dx = 2 sec 2 d, 4 + x 2 = 4 sec 2 , while x = 0 = = 0 , x = 2 = = 4 . In this case, I = 6 integraldisplay / 4 tan 2 sec 2 sec 2 d = 6 integraldisplay / 4 tan 2 d . But tan 2 = sec 2 - 1, so I = 6 integraldisplay / 4 ( sec 2 - 1 ) d = 6 bracketleftBig tan - bracketrightBig / 4 . gutierrez (ig3472) HW08 Neitzke (56585) 3 Consequently I = 3 2 (4- ) . 005 10.0 points Determine the indefinite integral I = integraldisplay 1- 2 x x 2- 1 dx ....
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HW 08 solution - gutierrez (ig3472) HW08 Neitzke (56585) 1...

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