Course:MATH101/2012-2013/210/WebWork Solutions/Assignment 6

Problem 10

Match the following improper integral with the improper integral below in which you can compare using the Comparison Test. Then determine whether the integral converge or diverge.

A. $\int _{1}^{\infty }{\frac {dx}{x}}$, B. $\int _{1}^{\infty }{\frac {dx}{x^{2}}}$, C. $\int _{1}^{\infty }{\frac {dx}{x^{3}}}$, and D. $\int _{1}^{\infty }{\frac {dx}{\sqrt {x}}}$

1. $\int _{1}^{\infty }{\frac {dx}{x^{2}+5}}$, Answer: B, Converges.

Key step: ${\frac {1}{x^{2}+5}}\leq {\frac {1}{x^{2}}}$

2. $\int _{1}^{\infty }{\frac {2+e^{-x}}{\sqrt {x}}}dx$, 'Answer: D, Diverges.

Key step: $e^{-x}>0~$, so ${\frac {2+e^{-x}}{\sqrt {x}}}>{\frac {1}{\sqrt {x}}}$

3. $\int _{1}^{\infty }{\frac {x}{x^{2}-0.5}}dx$, 'Answer: A, Diverges.

Key step: When the denominator is decreased, the fraction increases. So ${\frac {x}{x^{2}-0.5}}>{\frac {x}{x^{2}}}={\frac {1}{x}}$

4. $\int _{1}^{\infty }{\frac {2+\cos x}{x}}dx$, 'Answer: A, Diverges.

Key step: Recall the range of $\cos x$ is [-1,1], so $2+\cos x\geq 1$