Spring 2006 - Nagy's Class - Exam 2

# Spring 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 May 26, 2006 No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each question carefully. If any question is not clear, ask for clarification. Answer each question completely, and show all your work. 1. Evaluate the following integrals, (a) (8 points) Z x ln x 2 dx , (b) (8 points) Z ( x 2 - 3 x ) e x dx . (a) The integral above is a particular case of Z x ln( ax ) dx for a = 1 / 2. Integrating by parts, Z x ln( ax ) dx = x 2 2 ln( ax ) - Z x 2 2 1 ax adx , = x 2 2 ln( ax ) - 1 2 Z x dx, = x 2 2 ln( ax ) - x 2 4 + c, = x 2 2 ln( ax ) - 1 2 + c. Therefore, the solution is Z x ln x 2 dx = x 2 2 ln x 2 - 1 2 + c . (b) The integral above is a particular case of Z ( x 2 - ax ) e x dx for a = 3. Integrating twice by parts, Z ( x 2 - ax ) e x dx = ( x 2 - ax ) e x - Z (2 x - a ) e x dx, = ( x 2 - ax ) e x - (2 x - a ) e x - Z 2 e x dx , = ( x 2 - ax ) e x - (2 x - a ) e x + 2 e x + c, = x 2 - (2 + a ) x + (2 + a ) e x + c.

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