# MAT136H1 Lecture Notes - List Of Trigonometric Identities

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7.2 Integration Techniques
Trigonometric Integrals
Question #2 (Medium): Evaluating the Integral of Sine & Cosine With Even Power
Strategy
When  and or  are raised to even power, half-angle identities are helpful in reducing the even
power by half every time it is applied. Then the reduced power, in conjunction with trigonometric
identities, the integral can be simplified and evaluated.
Sample Question
Evaluate the integral.


Solution
Since both  and  are raised to even power, no single term can be isolated. By applying half-
angle identities, even power is reduced by half every time it is applied. So here: 



   

  
   
    
Then the three components in the integral can be evaluated separately.
So:



First:

  

  
 
   
Second:

  
  . Let , then   
Substituting into the integral,
  
  
 
  

  
Last component:


  
  
     
   
  
    

 
 
  
  
   .
Put all three pieces together:



 
  
 
 
  
  
   


 
  
  
Therefore, the integral is evaluated as:



 
 
 
  
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## Document Summary

Question #2 (medium): evaluating the integral of sine & cosine with even power. When and or are raised to even power, half-angle identities are helpful in reducing the even power by half every time it is applied. Then the reduced power, in conjunction with trigonometric identities, the integral can be simplified and evaluated. Since both and are raised to even power, no single term can be isolated. By applying half- angle identities, even power is reduced by half every time it is applied. Then the three components in the integral can be evaluated separately.

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