There really isn’t all that much to this problem other than use the formula given in the notes for this section.

\[{f_{{\rm{avg}}}} = \frac{1}{{\pi  - \left( { - \frac{\pi }{2}} \right)}}\int_{{ - \,\frac{\pi }{2}}}^{\pi }{{\cos \left( {2x} \right) - \sin \left( {\frac{x}{2}} \right)\,dx}} = \frac{2}{{3\pi }}\left. {\left( {\frac{1}{2}\sin \left( {2x} \right) + 2\cos \left( {\frac{x}{2}} \right)} \right)} \right|_{ - \,\frac{\pi }{2}}^\pi = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{2\sqrt 2 }}{{3\pi }}}}\]

Note that we are assuming your integration skills are pretty good at this point and won’t be showing many details of the actual integration process. This includes not showing either of the substitutions needed for the integral (you did catch both of them correct?).