10.1.24 Integral of a complex variable

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{e^{z}}{\sinh 2 z} d z \text {, where } \Omega=\left\{-\frac{\pi}{4}<\operatorname{Im} z<\frac{\pi}{4}\right\} \text {. } \]