If you want to study, imo, do lots of problems. The more you do, the more you get a feel of what to look for on each one.
Some general ones: 1. Draw the region described. 2. Find the number of poles, if any, and the order of each. 3. Compare them to the given region. Which ones lie in it? Which ones don't? 4. Then just apply Cauchy Residue Theorem.
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