\begin{align*} E[X] &= \int_{-\infty}^{\infty} x \cdot \frac{1}{\sigma \sqrt{2 \pi}} e^{ - \left ( \frac{x-\mu}{\sigma} \right ) ^2 / 2} dx \\ &= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} (\mu + z\sigma) \cdot e^{ -z^2 / 2} dz\\ &= \mu \cdot \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{ -z^2 / 2} dz + \sigma \cdot \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} z \cdot e^{ -z^2 / 2} dz\\ &= \mu \cdot 1 + \sigma \cdot 0\\ & = \mu \end{align*}