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!):

$\displaystyle

p(x;\lambda) =\left\{\begin{array}{cc}2 \lambda x\ e^{-\lambda{x^2}}&\mbox{ if } x\geq 0\\0 & \mbox{ if } x<0\end{array}\right. (\lambda>0)

$

I know that the even moments are:

$\displaystyle

E[x^{2n}]=\frac{n!}{\lambda^n}\ (n=0,1,2, ...)

$

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

$\displaystyle

M_X(t)=\int_0^\infty 2 \lambda x\ e^{-\lambda{x^2}+tx} dx\

$

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...