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227_sample_exam_solutions_spring_2009[1]

227_sample_exam_solutions_spring_2009[1] - Math 227.04...

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Math 227.04 Sample Exam 1 Solutions You must show your work in order to receive full credit. You may use a scienti°c calculator and one page double-sided sheet of notes. 1. (10 points) Transform the de°nite integral R 1 0 e 2 x p 1 + e 4 x dx using the substitution u = e 2 x : Make sure to transform the limits of integration. You are not required to evaluate the °nal integral. Solution: u = e 2 x ) du = 2 e 2 x dx , so e 2 x dx = 1 2 du: Also, e 4 x = ( e 2 x ) 2 = u 2 : Thus Z 1 0 e 2 x p 1 + e 4 x dx = Z x =1 x =0 p 1 + u 2 du 2 = 1 2 Z e 1 p 1 + u 2 du: 2. (10 points each) Find each of the following. (a) R 6 x x 2 ° 9 dx Solution: 6 x x 2 ° 9 = 6 x ( x + 3)( x ° 3) = A x + 3 + B x ° 3 = ° 18 ° 3 ° 3 x + 3 + 18 3+3 x ° 3 = 3 x + 3 + 3 x ° 3 : Thus Z 6 x x 2 ° 9 dx = Z [ 3 x + 3 + 3 x ° 3 ] dx = 3 ln j x + 3 j + 3 ln j x ° 3 j + C = 3 ln ° ° x 2 ° 9 ° ° + C: (b) R 3 1 x 2 ln x dx Solution: apply parts: u = ln x; dv = x 2 dx du = 1 x dx; v = 1 3 x 3 to get Z 3 1 x 2 ln x dx = 1 3 x 3 ln x j 3 1 ° 1 3 Z 3 1 x 3 ± 1 x dx = 9 ln 3 ° 1 9 ± 3 3 ° 1 3 ² = 9 ln 3 ° 26 9 (c) R ° 0 j cos x j dx Solution:
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