Trigonometric Integrals

From Department of Mathematics at UTSA
Jump to navigation Jump to search

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

Powers of Sine and Cosine

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

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

Find .

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

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

Find .

There is an even power of . Substituting gives so


Example 2

Find .

Let so . Then


Example 3

Find .

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

Making the substitution so ,

More trigonometric combinations

For the integrals or or use the identities

Example 1

Find .

We can use the fact that , so

Now use the oddness property of to simplify

And now we can integrate

Example 2

Find: .

Using the identities

Then

Resources

Licensing

Content obtained and/or adapted from: