022 Exam 2 Sample A, Problem 3

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Find the antiderivative of

This problem requires two rules of integration. In particular, you need
Integration by substitution (u - sub): If   is a differentiable functions whose range is in the domain of , then
We also need the derivative of the natural log since we will recover natural log from integration:


Step 1:  
Use a u-substitution with This means , or . After substitution we have
Step 2:  
We can now take the integral remembering the special rule resulting in natural log:
Step 3:  
Now we need to substitute back into our original variables using our original substitution to find
Step 4:  
Since this integral is an indefinite integral, we have to remember to add a constant  at the end.
Final Answer:  

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