Moment generating functions

[kola77]

Hello everybody! I have encountered a problem regarding the aforementioned subject. I need to find a moment generating function for the following probability density:
f(x)=1/2*e^(-x/2) ;x>0

What I have managed to do so far is this:
Mx(t)=E(e^tX)= integral (0 to infinity) 1/2*e^(-x/2) * e^tx dx = 1/2 integral (zero to infinity) e^(tx-x/2)dx


From here on I just don't know how to solve the integral. Would appreciate any help on this matter immensely. Thanks in advance!

 

[galactus]

\(\displaystyle \frac{1}{2}\int_{0}^{\infty}e^{\frac{-x}{2}}e^{tx}dx\)

\(\displaystyle =\frac{1}{2}\int_{0}^{\infty}e^{tx-\frac{x}{2}}dx\)

This can be done with a u sub:

Let \(\displaystyle u=tx-\frac{x}{2}, \;\ du=(\frac{2t-1}{2})dx\)

Can you proceed?. Let me know what you get.