Bounded from above integral with exponential

Question

I am trying to bound from above the following integral $\int_{\left\{|u| \geq u_n\right\}} u^{4} e^{-\frac{u^{2}}{2n}} du$ where $u_n = 2 \sqrt{nlogn}$. Could you please give me any idea how I could bound it? Thank you very much.

Answer 1

By the Cauchy--Schwarz inequality, $$ 2\int_{u_n}^{+\infty}u^4e^{-u^2/2n}\,du\leq 2\biggl(\int_{u_n}^{+\infty}u^3e^{-u^2/2n}\,du\biggr)^{1/2}\biggl(\int_{u_n}^{+\infty}u^5e^{-u^2/2n}\,du\biggr)^{1/2} $$ The integrals to the right can be calculated exactly by calculus methods. If I did it correctly, the order of the integral will be $\sqrt{n}(\ln n)^3$, which agrees with the order of your integral for large $n$.