9.2.13 Integrals depending on a parameter

Problem: Prove that the integral \( I(\alpha) \) converges uniformly on the set \( \mathrm{E} \), if: \[ I(\alpha)=\int_{0}^{+\infty} \cos x^{\alpha} d x, \mathrm{E}=\left[\alpha_{0} ;+\infty\right), \alpha_{0}>1 . \]