009B Sample Final 3, Problem 5

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Find the following integrals.

(a)  

(b)  

Foundations:  
1. Integration by parts tells us that
       
2. Since    we have
       


Solution:

(a)

Step 1:  
To calculate this integral, we use integration by parts.
Let    and  
Then,    and  
Therefore, we have
       
Step 2:  
Then, we integrate to get
       

(b)

Step 1:  
First, we use the identity    to get
       
Step 2:  
Now, we use  -substitution.
Let   
Then,    and  
Therefore, we have
       


Final Answer:  
   (a)    
   (b)    

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