10.1.26 Integral of a complex variable

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{d z}{\left(z^{4}+1\right) \sqrt{z^{2}+1}} d z \] where \( \Omega=\left\{\operatorname{Re} z>(\operatorname{Im} z)^{2}\right\} \), and \( \left.\sqrt{z^{2}+1}\right|_{z=0}=1 \).