Moment generating function

Ok, let's use the brute approach. This is a continuation of my previuos post:

http://www.mathhelpforum.com/math-he...ble-check.html

which is meant to analytically calculate the moments of the following pdf (this is my problem!):

I know that the even moments are:

In that previous post I'm using (unsuccesfully) the moment generating function:

You can see there why it does not take me to the expecetd results.

How can I get to the expected results?? Perhaps developing with Taylor??

I am required NOT to use higher level tools, like Laplace/Fourier, etc...