# HW8 - momin (rrm497) Homework 08 cheng (58520) 1 This...

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Unformatted text preview: momin (rrm497) Homework 08 cheng (58520) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the indefinite integral I = integraldisplay 1 x 2 6 x 7 dx 1. I = sin- 1 parenleftBig x 3 4 parenrightBig + C 2. I = ln vextendsingle vextendsingle vextendsingle x + 3 + radicalbig x 2 + 6 x 7 vextendsingle vextendsingle vextendsingle + C 3. I = sin- 1 parenleftBig x 4 3 parenrightBig + C 4. I = ln vextendsingle vextendsingle vextendsingle x + 3 + radicalbig x 2 6 x 7 vextendsingle vextendsingle vextendsingle + C 5. I = ln vextendsingle vextendsingle vextendsingle x 3 + radicalbig x 2 6 x 7 vextendsingle vextendsingle vextendsingle + C correct 6. I = ln vextendsingle vextendsingle vextendsingle x 3 + radicalbig x 2 + 6 x 7 vextendsingle vextendsingle vextendsingle + C Explanation: By completing the square we see that x 2 6 x 7 = ( x 2 6 x + 9 ) 16 = ( x 3) 2 16 . This suggests the substitution x 3 = 4 sec , for then dx = 4 sec tan d , while ( x 3) 2 16 = 16tan 2 . In this case I = integraldisplay 4 sec tan 4 tan d = integraldisplay sec d = ln | sec + tan | + C . Now x 3 = 4 sec = tan = radicalbig ( x 3) 2 16 4 , so I = ln vextendsingle vextendsingle vextendsingle x 3 + radicalbig ( x 3) 2 16 4 vextendsingle vextendsingle vextendsingle + C . Consequently I = ln vextendsingle vextendsingle vextendsingle x 3 + radicalbig x 2 6 x 7 vextendsingle vextendsingle vextendsingle + C . 002 10.0 points Evaluate the definite integral I = integraldisplay 2 1 x 2 + 5 x + 1 dx . Correct answer: 2 . 93279. Explanation: After division x 2 + 5 x + 1 = ( x 2 1) + 6 x + 1 = x 2 1 x + 1 + 6 x + 1 = x 1 + 6 x + 1 . In this case I = integraldisplay 2 1 parenleftBig x 1 + 6 x + 1 parenrightBig dx = bracketleftBig 1 2 x 2 x + 6 ln | x + 1 | bracketrightBig 2 1 = parenleftBig 1 1 2 parenrightBig + 6 parenleftBig ln 3 ln 2 parenrightBig . Consequently, I = 1 2 + 6 ln 3 2 = 2 . 93279 . momin (rrm497) Homework 08 cheng (58520) 2 003 10.0 points Evaluate the integral I = integraldisplay / 4 sec 2 x { 1 4 sin x } dx . 1. I = 3 4 2 2. I = 5 + 4 2 3. I = 3 2 2 4. I = 5 + 2 2 5. I = 3 + 2 2 6. I = 5 4 2 correct Explanation: Since sec 2 x { 1 4 sin x } = sec 2 x 4 sec x parenleftBig sin x cos x parenrightBig , we see that I = integraldisplay / 4 { sec 2 x 4 sec tan x } dx . But d dx tan x = sec 2 x , while d dx sec x = sec x tan x ....
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## This note was uploaded on 12/11/2009 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas at Austin.

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HW8 - momin (rrm497) Homework 08 cheng (58520) 1 This...

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