\displaystyle \mathbb{E}[X^3] =\int_{-\infty}^{+\infty} x^3\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}dx= \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^{+\infty} x^3e^{\frac{-(x-\mu)^2}{2\sigma^2}}dx