Any idea on how to integrate the following?

[Amannequin]

Homework Statement

The c.d.f. of a random variable X is given by the following formula

F(x) = 1 - e^-x2 x≥0

Find M(t). Moment generating function.

The Attempt at a Solution

I've found the pdf as

f(x) = F'(x) = 2xe^-x2 x≥0

To determine M(t) I need to do the following

M(t) = E(e^tX) = ∫e^tx 2xe^-x2

Then I'm not sure how to go about integrating. Any ideas or clues to point me in the right direction will be greatly appreciated. Thanks.

 

[jbunniii]

Try completing the square: $x^2 - tx = x^2 - tx + (t/2)^2 - (t/2)^2 = (x - t/2)^2 - (t/2)^2$, so
$$\int e^{tx}2xe^{-x^2}dx = e^{(t/2)^2} \int 2x e^{-(x-t/2)^2} dx$$
which should be manageable with a bit more work.