Winter 2006 - Nagy's Class - Exam 2

# Winter 2006 - Nagy's Class - Exam 2 - Name TA Name Math 20B...

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Name: Section Number: TA Name: Section Time: Math 20B. Midterm Exam 2 March 1, 2006 Turn off and put away your cell phone. No calculators or any other devices are allowed on this exam. You may use one page of notes, but no books or other assistance on this exam. Read each question carefully, answer each question completely, and show all of your work. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. 1. (6 points) Evaluate Z e i 2 x cos( x ) dx . You may leave the result in exponential form. Z e i 2 x cos( x ) dx = 1 2 Z e i 2 x ( e ix + e - ix ) dx, = 1 2 Z ( e i 3 x + e ix ) dx, = 1 2 1 3 i e i 3 x + 1 i e ix , = - i 6 ( e i 3 x + 3 e ix ) .

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2. (6 points) Evaluate Z 1 ( x 2 + 16) 2 dx . Introduce the substitution x = 4 tan( θ ), then dx = 4 sec 2 ( θ ) , and x 2 + 4 2 = 4 sec 2 ( θ ). Therefore, Z dx ( x 2 + 4 2 ) 2 = Z 1 (4 2 sec 2 ( θ )) 2 4 sec 2 ( θ ) dθ, = 1 4 3 Z sec 2 ( θ ) , = 1 4 3 Z cos 2 ( θ ) dθ, = 1 (2)4 3 Z [1 + cos(2 θ )] dθ, = 1 2 7 θ + 1 2 sin(2 θ ) + c, = 1 2 7 ( θ + sin( θ ) cos( θ )) + c.
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