007B Sample Midterm 2, Problem 5 Detailed Solution

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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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