\begin{align*} M(t) & = \int_{-\infty}^{\infty} e^{tx} \frac{1}{\sigma \sqrt{2 \pi}} e^{ -\left ( \frac{x-\mu}{\sigma} \right ) ^2 / 2} dx\\ & = \int_{-\infty}^{\infty} e^{t(z \sigma + \mu)} \frac{1}{\sqrt{2 \pi}} e^{ -z ^2 / 2} dz\\ & = e^{t \mu} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -(z ^2 - 2t z \sigma ) / 2} dz\\ & = = e^{t \mu} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -(z ^2 - 2t z \sigma + t^2 \sigma^2 - t^2 \sigma^2 ) / 2} dz\\ & = e^{t \mu+\frac{1}{2}t^2\sigma^2} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -\left (z - t \sigma \right )^2 / 2} dz\\ & = e^{t \mu+\frac{1}{2}t^2\sigma^2} \cdot 1 \end{align*}