Second moment and MGF of a Gaussian

Let $X\sim N(\mu,\sigma^2)$. Find $\mathbb{E}[X^2]$ and, for a constant $\lambda$, $\mathbb{E}[e^{\lambda X}]$.

$\mathbb{E}[X^2]=\mu^2+\sigma^2$ and $\mathbb{E}[e^{\lambda X}]=e^{\lambda\mu+\lambda^2\sigma^2/2}$. The first is $\operatorname{Var}(X)+(\mathbb{E}X)^2$. For the second, complete the square in the Gaussian integral; this is the moment generating function.