Difference between revisions of "022 Exam 2 Sample A, Problem 3"
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!Step 2:  !Step 2:  
    
−  We can now take the integral remembering the special rule:  +  We can now take the integral remembering the special rule resulting in natural log: 
    
    
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!Step 3:  !Step 3:  
    
−   Now we need to substitute back into our original variables using our original substitution <math style="verticalalign: 5%">u = 3x + 2</math>  +   Now we need to substitute back into our original variables using our original substitution <math style="verticalalign: 5%">u = 3x + 2</math> to find 
    
−    +   
+  ::<math>\frac{\log(u)}{3} = \frac{\log(3x + 2)}{3}.</math>  
}  }  
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 usubstitution 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: 

