Difference between revisions of "007B Sample Midterm 2, Problem 5 Detailed Solution"

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(Created page with "<span class="exam"> Evaluate the integral: ::<math>\int \frac{4x}{(x+1)(x^2+1)} ~dx</math> <hr> {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Backgroun...")
 
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>-2 \ln |x+1|+\ln |x^2+1|+2\arctan(x)+C</math>
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>-2 \ln |x+1|+\ln |x^2+1|+2\arctan(x)+C</math>
 
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[[007B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
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[[007B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 15:14, 23 November 2017

Evaluate the integral:


Background Information:  
Through partial fraction decomposition, we can write
       
for some constants


Solution:

Step 1:  
We need to use partial fraction decomposition for this integral.
To start, we let
       
Multiplying both sides of the last equation by  
we get
       
Step 2:  
If we let    the last equation becomes    So,  
If we let    then we get    Thus,  
Finally, if we let    we get   
Plugging in    and    we get  
So, in summation, we have
       
Step 3:  
Now, we have

       

For the remaining integrals, we use  -substitution.
For the first integral, we substitute  
For the second integral, the substitution is  
Then, we integrate to get

       


Final Answer:  
       

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