The question is let $I=\int_{\gamma(0,1)} \frac{Rz}{2z-i} dz$ and $J=\int_0^{2\pi} \frac{\cos^2(\theta)}{5-4\sin(\theta)} d\theta$. The problem wanted me to evaluate I which I did as $-\frac{\pi i}{4}$ then plug in $z=e^{i\theta}$ into $I$ and get $J$.
I plugged in to get $$i \int_0^{2\pi} \frac{\cos^2\theta + i\sin\theta \cos\theta}{2\cos\theta + 2i\sin\theta - i} d\theta$$ but I'm not sure how to proceed. I know from using online graphing that this comes out to $\frac{\pi}{4}$ but I'm not seeing the evaluation or even how to get it close to $J$.
Any help is much appreciated!