= Solution
For a counterclockwise simple contour containing $z_0$, and $f$ analytic on a neighbourhood of its interior and boundary, the <Cauchy integral formula> is
$$
\boxed{\frac1{2\pi i}\oint_C\frac{f(z)}{z-z_0}dz=f(z_0).}
$$
For the required rational integral, use <partial fraction decomposition>:
$$
\frac1{z^2-1}=\frac12\left(\frac1{z-1}-\frac1{z+1}\right).
$$
Both points $1,-1$ lie inside $|z|=2$. Applying the formula with the constant analytic function one gives $2\pi i$ for each of the two terms. They cancel, so
$$
\boxed{\oint_{|z|=2}\frac{dz}{z^2-1}=\frac12(2\pi i-2\pi i)=0.}
$$
Reversing the circle's orientation would leave this zero result unchanged.
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