Calculus/Integration techniques/Trigonometric Integrals
When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.
Powers of Sine and Cosine
editWe 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 .
Example
editFind .
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 .
Example
editFind .
As and we have
and expanding, the integrand becomes
Using the multiple angle identities
TODO: CORRECT FORMULA
then we obtain on evaluating
Powers of Tan and Secant
editTo evaluate .
- If is even and then substitute and use the identity .
- If and are both odd then substitute and use the identity .
- 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 1
editFind .
There is an even power of . Substituting gives so
Example 2
editFind .
Let so . Then
Example 3
editFind .
The trick to do this is to multiply and divide by the same thing like this:
Making the substitution so ,
More trigonometric combinations
editFor the integrals or or use the identities
Example 1
editFind .
We can use the fact that , so
Now use the oddness property of to simplify
And now we can integrate
Example 2
editFind: .
Using the identities
Then