Element of $L^p(\mathbb{R}^d)$

Question

Let $1\leq p \leq \infty$. Is there an easy way to argue that $e^{-\vert x\vert ^2}$ is an element of $L^p(\mathbb{R}^d)$ (integral w.r.t. $d$-dimensional Lebesgue measure)?

Answer 1

$$\int e^{-|x|^2}=\int\prod_{i=1}^de^{-x_i^2}=\prod_{i=1}^d\int e^{-x_i^2},$$ and $x_i\longmapsto e^{-x_i^2}$ is $L^p$ for all $i$