9.2.6 Integrals depending on a parameter

Problem: Prove that the integral \( \mathrm{I}(\alpha) \) converges uniformly on the set \( \mathrm{E} \), if: \[ I(a)=\int_{2}^{+\infty} x^{\alpha} e^{-2 x^{\alpha}} d x, \quad \mathrm{E}=[1 ; 2] . \]