009B Sample Final 1, Problem 4

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Compute the following integrals.




1. Integration by parts tells us that .
2. Through partial fraction decomposition, we can write the fraction    for some constants .
3. We have the Pythagorean identity .



Step 1:  
We first distribute to get
Now, for the first integral on the right hand side of the last equation, we use integration by parts.
Let and . Then, and .
So, we have
Step 2:  
Now, for the one remaining integral, we use -substitution.
Let . Then, .
So, we have


Step 1:  
First, we add and subtract from the numerator.
So, we have
Step 2:  
Now, we need to use partial fraction decomposition for the second integral.
Since , we let .
Multiplying both sides of the last equation by ,
we get .
If we let , the last equation becomes .
If we let , then we get  . Thus, .
So, in summation, we have  .
Step 3:  
If we plug in the last equation from Step 2 into our final integral in Step 1, we have
Step 4:  
For the final remaining integral, we use -substitution.
Let . Then, and  .
Thus, our final integral becomes
Therefore, the final answer is


Step 1:  
First, we write .
Using the identity , we get .
If we use this identity, we have
Step 2:  
Now, we proceed by -substitution. Let . Then, .
So we have
Final Answer:  

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