We have $$\begin{align}f(t) &= \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\cos(\omega)e^{j\omega t} \ dt \\ &=\frac{e^{j\omega t}}{2\pi(1-t^2)}\{jt\cos(\omega)+\sin(\omega)\}_{-\pi/2}^{\pi/2} \\ &=\frac{1}{\pi(1-t^2)}\cos\left(\frac{\pi t}{2}\right)\end{align}$$
How did we get the $jt\cos(\omega) + \sin(\omega)$ term?
Ultimately you're asking how to do $$\int \cos \omega\ e^{j \omega t} \, d\omega = \frac{e^{j \omega t}}{1-t^2} (jt \cos \omega + \sin \omega).$$ But this is just a standard application of integration by parts.