Finding the Moment Generating Function of $X^2$ when $X\sim N(0,1)$
Solution 1:
The mgf will only be defined when $t\lt 1/2$.
We need $$\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2(1/2-t)}\, dx.$$ The exponent can be written as $-(x^2/2)(1-2t)$.
Make the change of variable $u=x\sqrt{1-2t}$. Then $dx =\frac{1}{\sqrt{1-2t}}\,du$. Note that as $x$ travels from $-\infty$ to $\infty$, so does $u$. You should end up with something like $$\frac{1}{\sqrt{1-2t}}\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}}e^{-u^2/2}\,du.$$ Now we recognize that the integral part is $1$.