Problem. Prove that $$\int_{0}^{\infty}\frac{{\rm d}x}{x^{2}e^{x^{2}}}= \infty$$
Indeed, that's true because $$\int_{0}^{\infty}xe^{x^{2}}{\rm d}x= 1/2\int_{0}^{\infty}{\left ( x^{2} \right )}'e^{x^{2}}{\rm d}x= \infty$$ and there exists a numbers $x_{0}\geq 0$ so that $x_{0}e^{x_{0}^{2}}= 1,$ therefore $$\int_{0}^{\infty}\frac{{\rm d}x}{x^{2}e^{x^{2}}}\geq\int_{0}^{x_{0}}\frac{{\rm d}x}{x^{2}e^{x^{2}}}\geq\int_{0}^{x_{0}}\frac{{\rm d}x}{x}= \infty$$ I want to see any simplier solutions about my problem, that's my pleasure.
Edit, there are many complainings when I used $> \infty$ at the recent, and many of mathmaticians you can see here_ https://youtu.be/z1k8BUYmHz4, I want my equation clearer I don't do it on purpose