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Unformatted text preview: 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) R cos √ x +1 √ x +1 dx Solution: Let u = √ x + 1, then du = 1 2 √ x +1 dx . Thus, Z cos √ x + 1 √ x + 1 dx = Z 2 cos udu = 2 sin u + C = 2 sin √ x + 1 + C (b) R √ 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, Z p ln(2 x ) 2 ln(2 x ) x dx = Z ( √ 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) R 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 Z 200 x √ x 2 + 1 ( 1 + √ x 2 + 1 ) 101 dx = Z 200 u 101 du = 2 u 100 + C = 2 ( 1 + √ x 2 + 1 ) 100 + C 1 2. (10 points each) Evaluate the following definite integrals. Indicate the substitution2....
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This note was uploaded on 03/01/2009 for the course MATH 21B taught by Professor Vershynin during the Winter '08 term at UC Davis.
 Winter '08
 Vershynin

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