Complex integral by definition

by Lucas   Last Updated October 09, 2019 14:20 PM

Can anyone see a way to do the integral under the curve $\gamma(t)=2e^{it}, t\in[0, 2\pi]$:

$$\int _\gamma \frac{2z}{z^2-2}dz$$

My intention is to use the definition (This exercise appears right after the definition of complex integral). After some accounts I arrived at:

$$\int _\gamma \frac{2z}{z^2-2}dz= \int_o^{2\pi}\frac{i(cos(2t)+isen(2t))}{2(cos(2t)+isen(2t))+1}dt=0+4i\int_0^{2\pi}\frac{2+cos(2t)}{5+4cos(2t)}dt$$

But calculating this last integral is being very difficult. Does anyone see an easier way to solve this?

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