Solution
= Solution
Integrate against the <standard normal density> by <completing the square>:
$$
\begin{aligned}
M_X(t)&=\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{tx-x^2/2}\,dx\\
&=e^{t^2/2}\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-(x-t)^2/2}\,dx=e^{t^2/2}.
\end{aligned}
$$
Thus the <moment-generating function of a standard normal variable> is
$$
\boxed{M_X(t)=e^{t^2/2}\quad(t\in\mathbb R).}
$$