9.2.1 Integrals depending on a parameter

Problem: Explore the integral for uniform convergence with respect to the parameter in the region \( E \) : \[ \begin{array}{l} \int_{0}^{\infty} \sin (\alpha \sinh x) d x, \\ E=\left[\frac{1}{2}, \infty\right), \quad \sinh x=\frac{e^{x}-e^{-x}}{2} . \end{array} \]