even and odd functions this is the final topic

Question

Even and Odd Functions : This is the final topic that we have to discuss in this chapter. 

Firstly, an even function is any function which satisfies,

                                                                f ( x ) = x²

Typical examples of even functions are following,

f ( x ) = f ( x )                                  f ( x ) = cos (x )

An odd function is any function that satisfies,

f (- x ) = - f ( x )

The typical examples of odd functions are following,

f ( x ) = x³                           f (x ) = sin ( x )

There are some nice facts regarding integrating even & odd functions over the interval [-a,a]. If f(x) is an even function then,

∫^a _(-a)      f ( x ) dx = 2∫^a₀ f ( x ) dx

Similarly, if f(x) is an odd function then,

 ∫^a _(-a)     f ( x ) dx =0

Note as well that in order to use these facts the limit of integration has to be the same number, however opposite signs!