Here is my attempt (without using residue theorem or Cauchy, just basic stuff):
$$\int_{\gamma}^{} \frac{1}{z^2-1}dz=\frac{1}{2}\Big( \int_{\gamma}^{} \frac{1}{z-1}dz-\int_{\gamma}{}\frac{1}{z+1}dz\Big)=\frac{1}{2}\Big( \int_{\gamma}^{} \frac{1}{z-1}dz+\int_{-\gamma}{}\frac{1}{z+1}dz\Big)=\frac{1}{2}\Big( \int_{0}^{2\pi} \frac{ie^{it}}{1+e^{it}-1}dt+\int_{0}^{2\pi}\frac{-ie^{it}}{-1-e^{it}+1}dt\Big)= \int_{0}^{2\pi} \ idt=2\pi i$$
Is this correct? Any feedback is appreciated. Thank you in advance.