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.

Powers of Sine and Cosine[edit]

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 .

Example[edit]

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 .

Example[edit]

Find .

As and we have

and expanding, the integrand becomes

Using the multiple angle identities

then we obtain on evaluating

Powers of Tan and Secant[edit]

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 1[edit]

Find .

There is an even power of . Substituting gives so


Example 2[edit]

Find .

Let so . Then


Example 3[edit]

Find .

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

Making the substitution so ,

More trigonometric combinations[edit]

For the integrals or or use the identities

Example 1[edit]

Find .

We can use the fact that , so

Now use the oddness property of to simplify

And now we can integrate


Example 2[edit]

Find: .

Using the identities

Then

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Integration techniques/Trigonometric Integrals