007B Sample Midterm 3, Problem 5 Detailed Solution

From Math Wiki
Jump to navigation Jump to search

Evaluate the following integrals.



Background Information:  
1. Integration by parts tells us that
2. Through partial fraction decomposition, we can write the fraction
       for some constants



Step 1:  
We proceed using integration by parts.
Let    and  
Then,    and  
Therefore, we have
Step 2:  
We integrate to get


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


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


Final Answer:  

Return to Sample Exam