Chapter 14.11, Problem 30MP

Show that $PV{\displaystyle {\int }_0^{\infty }\frac{\cos (\ln x)}{x^2+1}}dx=\frac{\pi }{2\cosh (\pi /2)}$ by integrating $e^{i\ln z}/\left(z^2-1\right)$ around a contour like Figure 7.3 but rotated ${90}^{°}$ clockwise so the straight side is along the $y$ axis.