Difference between revisions of "022 Exam 2 Sample A, Problem 3"

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!Step 2:  
 
!Step 2:  
 
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|We can now take the integral remembering the special rule:
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|We can now take the integral remembering the special rule resulting in natural log:
 
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!Step 3:  
 
!Step 3:  
 
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| Now we need to substitute back into our original variables using our original substitution <math style="vertical-align: -5%">u = 3x + 2</math>
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| Now we need to substitute back into our original variables using our original substitution <math style="vertical-align: -5%">u = 3x + 2</math> to find
 
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| to find&nbsp; <math>\frac{\log(u)}{3} = \frac{\log(3x + 2)}{3}.</math>
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::<math>\frac{\log(u)}{3} = \frac{\log(3x + 2)}{3}.</math>
 
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Latest revision as of 07:47, 16 May 2015

Find the antiderivative of


Foundations:  
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:

 Solution:

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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