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test1ans - Millersville University Department of...

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Millersville University Name Answer Key Department of Mathematics MATH 211, Calculus II , Test 1 July 28, 2009 Please answer the following questions. Your answers will be evaluated on their correctness, completeness, and use of mathematical concepts we have covered. Please show all work and write out your work neatly. Answers without supporting work will receive no credit. The point values of the problems are listed in parentheses. 1. (12 points) Evaluate the following indefinite integral. x 4 ln x dx Integrate by parts letting u = ln x du = 1 x dx v = 1 5 x 5 dv = x 4 dx then x 4 ln x dx = 1 5 x 5 ln x - 1 5 x 5 1 x dx = 1 5 x 5 ln x - 1 5 x 4 dx = 1 5 x 5 ln x - 1 25 x 5 + C.
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2. (13 points) Evaluate the following indefinite integral. 1 x 3 + x dx Since the integrand is a rational function, perform a partial fraction decomposition. 1 x 3 + x = 1 x ( x 2 + 1) = A x + Bx + C x 2 + 1 1 = A ( x 2 + 1) + ( Bx + C ) x Let x = 0: 1 = A (0 2 + 1) + ( B (0) + C )(0) = A 1 = 1( x 2 + 1) + ( Bx + C ) x 0 = x 2 + ( Bx + C ) x 0 = x + Bx + C Let x = 0: 0 = 0 + B (0) + C = C 0 = x + Bx + 0 0 = x + Bx 0 = 1 + B = B = - 1 Thus 1 x 3 + x dx = 1 x - x x 2 + 1 dx = 1 x dx - x x 2 + 1 dx = ln | x | -
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