Difference between revisions of "009B Sample Final 2, Problem 7"

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(Replaced content with "<span class="exam">Evaluate the following integrals or show that they are divergent: <span class="exam">(a)  <math>\int_1^\infty \frac{\ln x}{x^4}~dx</math> <span c...")
 
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<span class="exam">(b) &nbsp;<math> \int_0^1 \frac{3\ln x}{\sqrt{x}}~dx</math>
 
<span class="exam">(b) &nbsp;<math> \int_0^1 \frac{3\ln x}{\sqrt{x}}~dx</math>
  
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!Foundations: &nbsp;
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[[009B Sample Final 2, Problem 7 Solution|'''<u>Solution</u>''']]
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|'''1.''' How could you write &nbsp; <math style="vertical-align: -14px">\int_0^{\infty} f(x)~dx</math> so that you can integrate?
 
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&nbsp; &nbsp; &nbsp; &nbsp; You can write &nbsp; <math>\int_0^{\infty} f(x)~dx=\lim_{a\rightarrow\infty} \int_0^a f(x)~dx.</math>
 
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|'''2.''' How could you write &nbsp; <math>\int_{0}^1 \frac{1}{x}~dx?</math>
 
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&nbsp; &nbsp; &nbsp; &nbsp; The problem is that &nbsp;<math>\frac{1}{x}</math>&nbsp; is not continuous at &nbsp;<math style="vertical-align: 0px">x=0.</math>
 
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&nbsp; &nbsp; &nbsp; &nbsp; So, you can write &nbsp;<math style="vertical-align: -15px">\int_{0}^1 \frac{1}{x}~dx=\lim_{a\rightarrow 0} \int_{a}^1 \frac{1}{x}~dx.</math>
 
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'''Solution:'''
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[[009B Sample Final 2, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''(a)'''
 
  
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!Step 1: &nbsp;
 
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!Step 2: &nbsp;
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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!Final Answer: &nbsp;
 
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|'''(a)'''
 
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|'''(b)'''
 
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[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 17:33, 2 December 2017

Evaluate the following integrals or show that they are divergent:

(a)  

(b)  

Solution


Detailed Solution


Return to Sample Exam

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