MAT136H1 Lecture Notes - Product Rule

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7.1 Integration Techniques
Integration by Parts
Question #1 (Easy): Solving Indefinite Integral
Strategy
Integration by parts is useful when the integral contains two seemingly different function that cannot
further be simplified, so that:   .
This is after the form of the Product Rule. Assign simpler variables like    and let   ,
then by taking derivative of ,    and by taking the anti-derivative of ,  .
Sample Question
Evaluate the integral using integration by parts.
 

Solution
More complex function that simplifies when its derivative is taken should be assigned to . So here,
   . Then   
. Taking derivative of gives   , and taking the anti-
derivative of  gives  

 

.
Given the form of integration by parts:
     
Here:  
  

 

  . Since the second integral
has two functions that cannot be simplified, integration by parts needs to be taken one more time.
So again let   , and 

. Then,    and   
 
.
Then simplify the second integral:

  
 
 
 
  

 

  

 

  

 

  .
Merge this into the first part of the answer:  

  

 

 
 
  

 

  
  

 

  
Therefore, by integration by parts the integral is evaluated as:
 

  

 

  
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Document Summary

Integration by parts is useful when the integral contains two seemingly different function that cannot further be simplified, so that: ( ) ( ) ( ) ( ) ( ) ( ) . This is after the form of the product rule. Assign simpler variables like ( ) and let ( ) , then by taking derivative of , ( ) and by taking the anti-derivative of , ( ). More complex function that simplifies when its derivative is taken should be assigned to . Taking derivative of gives , and taking the anti- derivative of gives. Since the second integral has two functions that cannot be simplified, integration by parts needs to be taken one more time. Therefore, by integration by parts the integral is evaluated as:

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