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Possible values of $I := $ $\frac{1}{2\pi i}\int_\gamma \frac{2i}{z^2 + 1}dz$ where $\gamma $ is any closed curve in upper half plane not passing through $i$.

My approach:

There are two cases possible:

Case I: $i$ lies inside $\gamma$, then using residue formula we have $I = $ res$_i \big(\frac{2i}{(z+i)(z-i)}\big) = \frac{2i}{2i} = 1.$
Since orientation is not specified, $I = -1$ in case of opposite orientation.

Case II: $i$ doesn't lies inside $\gamma$, then $I = 0$.

But the answer given is set of all integers.

henceproved
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"Inside" and "outside" aren't quite precise enough for this. The curve can loop around $i$ as many times as you would like, and the residue formula has to take the winding number into account. Hence $\mathbb{Z}$ to account for all windings and both orientations.