Consider $ ∫_0^{2π} \frac{cos2θ}{5+4 cosθ} \ dθ =∫_0^{2π} \frac{e^{2iθ}}{5+4 cosθ} \ dθ$

To evaluate above integral, let $z = e^{iθ}$ ∴ $dz = ie^{iθ} \ dθ$ , $dθ=\frac{dz}{iz}$ & $cosθ= \frac{z^2+1}{2z}$ since θ varies from 0 to 2π therefore z moves around a unit circle |z|=1 , putting above values …