Calculus : Triginometric Integrals

Trigonometric Integrals

The object of this section is to work out how to do some types of integrals of trig functions, particularly where you have powers of trig functions, but also some others. There are three basic types that we will deal with:

Integrals of the Form

This situation we have three potential cases:

n is odd

If the power of cosine is odd, say , then we re-write the integral as:

and then recall that

, so the integral is

Now we substitute , which means that , and the integral becomes:

which is a polynomial and easy to evaluate.

m is odd

Both m and n are even.

Note that if both m and n are odd, you can use either of the first two methods.

Integrals of the Form

There are two cases which are easy to deal with, and one which is slightly harder:

n is even

m is odd (and n is at least 1)

m is even

You can use the same sort of trick with trigonometric identities to evaluate integrals of the form [Maple Math] .

Integrals of the Form , or

These types of integrals are extremely important, because they often occur when working with Fourier series (an important mathematical tool used widely in Engineering applications).

Fortunately they are not difficult to deal with, simply use the appropriate trig identities:

These are then really easy to integrate.