MAT21B_07Exam1Key

MAT21B_07Exam1Key - Solution to MAT21B Midterm I Dr Qinglan...

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Solution to MAT21B Midterm I Dr. Qinglan Xia Oct.19, 2007 Time: 12:10–1:00pm Name Student ID Instructions. This is a closed book/notes/friends exam. Totally 4 pages, 6 problems. No calculator is allowed. 1. (10 points each) Evaluate the following integrals. Indicate the substitution that you make (if you made one). (a) cos x +1 x +1 dx Solution: Let u = x + 1, then du = 1 2 x +1 dx . Thus, cos x + 1 x + 1 dx = 2 cos udu = 2 sin u + C = 2 sin x + 1 + C (b) ln(2 x ) - 2 ln(2 x ) x dx . Solution: Let u = ln (2 x ), then du = 1 2 x (2 x ) dx = 2 2 x dx = dx x . Thus, ln(2 x ) - 2 ln(2 x ) x dx = ( u - 2 u ) du = 2 3 u 3 2 - u 2 + C = 2 3 ln (2 x ) 3 2 - (ln (2 x )) 2 + C (c) 200 x x 2 +1 ( 1+ x 2 +1 ) 101 dx Solution: Let u = 1 + x 2 + 1, then du = x x 2 +1 dx and 200 x x 2 + 1 ( 1 + x 2 + 1 ) 101 dx = 200 u 101 du = - 2 u 100 + C = - 2 ( 1 + x 2 + 1 ) 100 + C 1

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2. (10 points each) Evaluate the following definite integrals. Indicate the substitution that you make (if you made one). (a) 1 0 e x +1 e 2 x dx Solution: 1 0 e x + 1 e 2 x dx = 1 0 e x e 2 x + 1 e 2 x dx = 1 0 e - x + e - 2 x dx = - e - x - e - 2 x 2 | x =1 x =0 = - 1 e - 1 2 e 2 - - 1 - 1 2 = - 1 e - 1 2 e 2 + 3
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