midterm2010soln

midterm2010soln - Solution to MAT21B Midterm 1 Dr. Qinglan...

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Unformatted text preview: Solution to MAT21B Midterm 1 Dr. Qinglan Xia Monday, April 19, 2010 Time: 2:103:00pm 1. (30 points) Evaluate the following integrals. Indicate the substitution that you make (if you made one). (a) R 6 t 2- 10 t- 2 t dt Solution: Z 6 t 2- 10 t- 2 t dt = Z 6 t- 10 t- 3 dt = 3 t 2 + 5 t- 2 + C. (b) R (2 x + 3 cos (3 x )) e 1+ x 2 +sin(3 x ) dx Solution: Let u = 1 + x 2 + sin(3 x ), then du = (2 x + 3cos(3 x )) dx. Thus, Z (2 x + 3cos(3 x )) e 1+ x 2 +sin(3 x ) dx = Z e u du = e u + C = e 1+ x 2 +sin(3 x ) + C. (c) R ( x +2)cos x 2 +4 x +2 x 2 +4 x +2 dx Solution: Let u = x 2 + 4 x + 2, then du = x +2 x 2 +4 x +2 dx . Thus, Z ( x + 2)cos x 2 + 4 x + 2 x 2 + 4 x + 2 dx = Z cos( u ) du = sin u + C = sin x 2 + 4 x + 2 + C. 2. (30 points) Evaluate the following definite integrals. Indicate the substitution that you make (if you made one). Justify your answers. (a) R 1 4 x 3 x 4 +1 dx Solution: Let u = x 4 + 1, then du = 4 x 3 dx. When x = 0, u = 1 and when x = 1,...
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midterm2010soln - Solution to MAT21B Midterm 1 Dr. Qinglan...

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