How to show that $\lim_{N\to\infty}\int_{|x|>N} |x|^3e^{-4x^2} dx=0$?

Question

I have to find the following limit: $$\lim_{N\to\infty}\int_{|x|>N} |x|^3e^{-4x^2} dx=0$$ The hard part is permutting limit and integral. I do not know how to apply the monotone convergence theorem since $|x|^3e^{-4x^2}$ is not $L^1(\mathbb{R})$.