q4solns - Mathematics 104, Quiz 4, Friday, March 7, 2008 1....

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Unformatted text preview: Mathematics 104, Quiz 4, Friday, March 7, 2008 1. Determine whether integraldisplay dx x 2 + 4 x + 9 converges or diverges. If it converges find the limit. integraldisplay dx x 2 + 4 x + 9 = integraldisplay dx ( x + 2) 2 + 5 = 1 5 arctan parenleftbigg x + 2 5 parenrightbigg . Since arctan( t ) goes to / 2 as t goes to infinity the original integral converges to 2 5- 1 5 arctan(2 / 5). 2. Consider integraldisplay x 1 + x 4 dx . Does this improper integral converge or diverge? Justify your answer using integration or one of the comparison tests. First note that the only problem is that we integrate on an infinite region. There are no vertical asymptotes. The function has no problem at x = 0 (or anywhere else in [0 , ). Next, we note that when x is large the integrand in asymptotic to 1 /x : lim x x 1 + x 4 / 1 x = lim x x 2 1 + x 4 = lim x radicalbigg x 4 1 + x 4 = 1 By the limit comparison test we conclude that integraldisplay...
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This note was uploaded on 04/07/2008 for the course MAT 104 taught by Professor Edwardnelson during the Spring '08 term at Princeton.

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