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| | ::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> | | ::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
| + | [[009B Sample Final 3, Problem 7 Solution|'''<u>Solution</u>''']] |
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| − | |'''Direct Comparison Test for Improper Integrals'''
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| − | | Let <math style="vertical-align: -5px">f</math> and <math style="vertical-align: -5px">g</math> be continuous on <math style="vertical-align: -5px">[a,\infty)</math>
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| − | | where <math style="vertical-align: -5px">0\le f(x)\le g(x)</math> for all <math style="vertical-align: 0px">x</math> in <math style="vertical-align: -5px">[a,\infty).</math>
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| − | | '''1.''' If <math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math> converges, then <math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math> converges.
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| − | | '''2.''' If <math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math> diverges, then <math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math> diverges.
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| − | '''Solution:''' | + | [[009B Sample Final 3, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | !Step 1:
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| − | !Step 2:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |}
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| | [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | | [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
