Difference between revisions of "009B Sample Final 3, Problem 7"

Revision as of 09:21, 1 March 2017

Does the following integral converge or diverge? Prove your answer!

\int _{1}^{\infty }{\frac {\sin ^{2}(x)}{x^{3}}}~dx

Foundations:
Direct Comparison Test for Improper Integrals
Let $f$ and $g$ be continuous on $[a,\infty )$
where $0\leq f(x)\leq g(x)$ for all $x$ in $[a,\infty ).$
1. If $\int _{a}^{\infty }g(x)~dx$ converges, then $\int _{a}^{\infty }f(x)~dx$ converges.
2. If $\int _{a}^{\infty }f(x)~dx$ diverges, then $\int _{a}^{\infty }g(x)~dx$ diverges.

Solution:

Final Answer:

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 !Foundations: &nbsp;
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+|'''Direct Comparison Test for Improper Integrals'''
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+|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g</math>&nbsp; be continuous on &nbsp;<math style="vertical-align: -5px">[a,\infty)</math>
 |-
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+|&nbsp; &nbsp; &nbsp; &nbsp; where &nbsp;<math style="vertical-align: -5px">0\le f(x)\le g(x)</math>&nbsp; for all &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; in &nbsp;<math style="vertical-align: -5px">[a,\infty).</math>
 |-
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+|&nbsp; &nbsp; &nbsp; &nbsp;'''1.'''&nbsp; If &nbsp;<math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math>&nbsp; converges, then &nbsp;<math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math>&nbsp; converges.
+|-
+|&nbsp; &nbsp; &nbsp; &nbsp;'''2.'''&nbsp; If &nbsp;<math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math>&nbsp; diverges, then &nbsp;<math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math>&nbsp; diverges.
 |}