007B Sample Midterm 2, Problem 2 Detailed Solution

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Evaluate

(a)  

(b)  


Background Information:  
How would you integrate  

        You can use  -substitution.

        Let  
        Then,  

        Thus,

       


Solution:

(a)

Step 1:  
We multiply the product inside the integral to get

       

Step 2:  
We integrate to get
      
We now evaluate to get

       

(b)

Step 1:  
We use  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} -substitution.
Let  
Then,    and  
Also, we need to change the bounds of integration.
Plugging in our values into the equation    we get
         and  
Therefore, the integral becomes
       
Step 2:  
We now have

       

Therefore,
       


Final Answer:  
    (a)    
    (b)    

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