First four raw moments of a standard normal

Let $X$ follow a standard normal distribution. Compute $\mathbb{E}[X]$, $\mathbb{E}[X^2]$, $\mathbb{E}[X^3]$ and $\mathbb{E}[X^4]$.

$0$, $1$, $0$ and $3$. Odd moments vanish because the density is symmetric. The MGF $e^{t^2/2}=1+\tfrac{t^2}{2}+\tfrac{t^4}{8}+\dots$ gives $\mathbb{E}[X^2]=2!\cdot\tfrac12=1$ and $\mathbb{E}[X^4]=4!\cdot\tfrac18=3$ (or integrate by parts: $\mathbb{E}[X^{n}]=(n-1)\mathbb{E}[X^{n-2}]$).