022 Exam 2 Sample A, Problem 3

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Find the antiderivative of


Foundations:  
This problem requires two rules of integration. In particular, you need
Integration by substitution (U - sub): If and are differentiable functions, then

    

The Product Rule: If and are differentiable functions, then

    

The Quotient Rule: If and are differentiable functions and  , then

    
Additionally, we will need our power rule for differentiation:
for ,
as well as the derivative of natural log:

 Solution:

Step 1:  
Use a U-substitution with This means , and after substitution we have
Step 2:  
We can now take the integral remembering the special rule:
Step 3:  
Since this integral is an indefinite integral we have to remember to add C at the end.
Final Answer:  

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