Calculus/Integration techniques/Trigonometric Integrals

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Integration techniques/Trigonometric Integrals

When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.

Contents

Powers of Sine and CosineEdit

We will give a general method to solve generally integrands of the form   . First let us work through an example.

 

Notice that the integrand contains an odd power of cos. So rewrite it as

 

We can solve this by making the substitution   so   . Then we can write the whole integrand in terms of   by using the identity

  .

So

   
 
 
 
 

This method works whenever there is an odd power of sine or cosine.

To evaluate   when either   or   is odd.

  • If   is odd substitute   and use the identity   .
  • If   is odd substitute   and use the identity   .

ExampleEdit

Find   .

As there is an odd power of   we let   so   . Notice that when   we have   and when   we have   .

   
 
 
 
 
 

When both   and   are even things get a little more complicated.

To evaluate   when both   and   are even.


Use the identities   and   .

ExampleEdit

Find   .

As   and   we have

 

and expanding, the integrand becomes

 

Using the multiple angle identities

   
 
 

then we obtain on evaluating

 

Powers of Tan and SecantEdit

To evaluate   .

  1. If   is even and   then substitute   and use the identity   .
  2. If   and   are both odd then substitute   and use the identity   .
  3. If   is odd and   is even then use the identity   and apply a reduction formula to integrate   , using the examples below to integrate when   .

Example 1Edit

Find   .

There is an even power of   . Substituting   gives   so

 


Example 2Edit

Find   .

Let   so   . Then

   
 
 
 
 


Example 3Edit

Find   .

The trick to do this is to multiply and divide by the same thing like this:

   
 

Making the substitution   so   ,

   
 
 

More trigonometric combinationsEdit

For the integrals   or   or   use the identities

  •  
  •  
  •  

Example 1Edit

Find   .

We can use the fact that   , so

 

Now use the oddness property of   to simplify

 

And now we can integrate

   
 


Example 2Edit

Find:  .

Using the identities

 

Then

   
 
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Integration techniques/Trigonometric Integrals