Integrating Even and Odd Functions

First, a few definitions:

A function is called even if f(x) = f(-x) for all values of x.

Geometrically, an even function is symmetric about the y axis.

A function is called odd if f(x) = -f(-x) for all values of x.

Geometrically, an odd function will be such that, if you rotate the graph 180 degrees (or pi radians), it will land on top of itself.

Now, the things you have to know:

If f(x) is an even function then,

If f(x) is an odd function then,

Example:

First notice that the numbers at the bottom and top of the integral (-3 and 3) are the same except for a sign change. And the function inside is very difficult to integrate. That should make you think about using the shortcuts I mentioned above.

Now, if we let , then let's see what we get when we replace x with -x:

(Squaring a negative number is the same as squaring the corrosponding positive number. Cubing a negative number is the same as taking the negative of the cube of the positive number)

So, f(x) is an odd function and we conclude that