cal8 - choi(kc24547 – HW08 – BERG –(56525 1 This...

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Unformatted text preview: choi (kc24547) – HW08 – BERG – (56525) 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 5 3 1 4 + ( x- 3) 2 dx . 1. I = 2 2. I = 1 4 π 3. I = 2 π 4. I = 1 4 5. I = 1 8 π correct 6. I = 1 2 π Explanation: Set 2 tan u = x- 3. Then 4 + ( x- 3) 2 = 4 + (2 tan u ) 2 = 4(1 + tan 2 u ) = 4 sec 2 u , while 2 sec 2 u du = dx . Also x = 3 = ⇒ u = 0 , and x = 5 = ⇒ u = π 4 . In this case I = integraldisplay π/ 4 2 sec 2 u 4 sec 2 u du = 1 2 integraldisplay π/ 4 du. Consequently, I = 1 2 bracketleftBig u bracketrightBig π/ 4 = 1 8 π . 002 10.0 points Evaluate the integral I = integraldisplay 1 / 2 sin- 1 x √ 1- x 2 dx . 1. I = π 2 72 correct 2. I = π 2 9 3. I = π 2 18 4. I = π 2 4 5. I = π 2 8 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 = π 6 . In this case I = integraldisplay π 6 u cos u cos u du = integraldisplay π 6 u du . Consequently, I = bracketleftbigg u 2 2 bracketrightbigg π 6 = π 2 72 . 003 10.0 points Determine the integral I = integraldisplay 1 ( x 2 + 4) 3 2 dx . choi (kc24547) – HW08 – BERG – (56525) 2 1. I = 1 4 √ x 2 + 4 + C 2. I = √ x 2 + 4 4 x + C 3. I = √ x 2 + 4 x + C 4. I = x √ x 2 + 4 + C 5. I = x √ x 2 + 4 4 + C 6. I = x 4 √ x 2 + 4 + C correct 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 1 3 x 2 1 + x 2 dx . 1. I = 3 4 ( π- 2) 2. I = 3 8 ( π + 2) 3. I = 3 4 (4- π ) correct 4. I = 3 4 (4 + π ) 5. I = 3 8 (4- π ) 6. I = 3 8 ( π- 2) Explanation: Let x = tan θ ; then dx = sec 2 θ dθ, 1 + x 2 = sec 2 θ , while x = 0 = ⇒ θ = 0 , x = 1 = ⇒ θ = π 4 . In this case, I = 3 integraldisplay π/ 4 tan 2 θ sec 2 θ sec 2 θ dθ = 3 integraldisplay π/ 4 tan 2 θ dθ . choi (kc24547) – HW08 – BERG – (56525) 3 But tan 2 θ = sec 2 θ- 1, so I = 3 integraldisplay π/ 4 ( sec 2 θ- 1 ) dθ = 3 bracketleftBig tan θ- θ bracketrightBig π/ 4 . Consequently I = 3 4 (4- π ) . 005 10.0 points Determine the indefinite integral I = integraldisplay 4- 3 x √ x 2- 1 dx ....
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas.

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cal8 - choi(kc24547 – HW08 – BERG –(56525 1 This...

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