022 Sample Final A, Problem 12

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Find the antiderivative:

This problem requires an advanced rule of integration, namely
Integration by substitution (u - sub): If   is a differentiable functions whose range is in the domain of , then


Step 1:  
Use a -substitution with This means , or  . After substitution, we have
Step 2:  
From what should be well-known property,
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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