022 Exam 2 Sample B, Problem 7

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Find the antiderivatives:


This problem requires Integration by substitution (u - sub): If   is a differentiable functions whose range is in the domain of , then
We also need our power rule for integration:
  for ,
as well as the convenient antiderivative:


(a) Step 1:  
(a) Use a u-substitution with This means , or . Substituting, we have
(a) Step 2:  
Now, we need to substitute back into our original variable using our original substitution
to find 
(a) Step 3:  
Since this integral is an indefinite integral, we have to remember to add a constant  at the end.
Unlike part (a), this requires no substitution. We can integrate term-by-term to find
Then, we evaluate:
Final Answer:  

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