CalcAnswers7 - nav277 Homework 7 Odell (58340) 1 This...

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Unformatted text preview: nav277 Homework 7 Odell (58340) 1 This print-out should have 19 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 2 3 x 2 4- x 2 dx . 1. I = 2- 3 2 2. I = 2 3- 1 3. I = 3 parenleftBig 2 3- 3 2 parenrightBig 4. I = 3 parenleftBig 2- 1 parenrightBig correct 5. I = 3 2 parenleftBig 3- 3 2 parenrightBig 6. I = 3 2 parenleftBig 2- 1 parenrightBig Explanation: Set x = 2 sin . Then dx = 2 cos d , while x = 0 = = 0 , x = 2 = = 4 . In this case I = integraldisplay / 4 12 sin 2 2 cos 2 cos d = 12 integraldisplay / 4 sin 2 d = 6 integraldisplay / 4 (1- cos 2 ) d = 6 bracketleftBig - 1 2 sin2 bracketrightBig / 4 . Consequently, I = 3 parenleftBig 2- 1 parenrightBig . keywords: 002 10.0 points Determine the integral I = integraldisplay x 2 (16- x 2 ) 3 / 2 dx . 1. I = 4 x 16- x 2- sin- 1 parenleftBig x 16 parenrightBig + C 2. I = 4 x 2 16- x 2 + sin- 1 parenleftBig x 2 16 parenrightBig + C 3. I = x 2 16- x 2 + sin- 1 parenleftBig x 2 4 parenrightBig + C 4. I = x 16- x 2- sin- 1 parenleftBig x 4 parenrightBig + C correct 5. I = 4 x 16- x 2- sin- 1 parenleftBig x 2 16 parenrightBig + C 6. I = x 16- x 2 + sin- 1 parenleftBig x 4 parenrightBig + C Explanation: Let x = 4 sin . Then dx = 4 cos d , 16- x 2 = 16cos 2 . In this case, I = integraldisplay 16 4 sin 2 cos 4 3 cos 3 d = integraldisplay sin 2 cos 2 d = integraldisplay tan 2 d . Now tan 2 = sec 2 - 1 , d d tan = sec 2 , nav277 Homework 7 Odell (58340) 2 and so I = integraldisplay (sec 2 - 1) d = tan - + C . Consequently, I = x 16- x 2- sin- 1 parenleftBig x 4 parenrightBig + C with C ann arbitrary constant. 003 10.0 points Determine the integral I = integraldisplay 2 ( x 2 + 4) 3 2 dx . 1. I = x x 2 + 4 + C 2. I = x 2 + 4 2 x + C 3. I = x 2 x 2 + 4 + C correct 4. I = x 2 + 4 x + C 5. I = x x 2 + 4 2 + C 6. I = 1 2 x 2 + 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 4 8 sec 2 u sec 3 u du = 1 2 integraldisplay cos u du , and so I = 1 2 sin u + C = 1 2 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 2 x 2 + 4 + C with C an arbitrary constant. keywords: trig substitution 004 10.0 points Determine the indefinite integral I = integraldisplay 3 + x x 2- 1 dx ....
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CalcAnswers7 - nav277 Homework 7 Odell (58340) 1 This...

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