Discussion
Put these steps of evaluating a real integral by residues in order. 1. Sum the residues of the poles in the upper half plane 2. Show the semicircular arc contributes nothing in the limit 3. Multiply by \( 2\pi i \) and take the real part if required 4. Replace the real integrand by a function of a complex variable and choose the contour
Answer: A
The complex integrand and contour come first. The arc estimate must be settled next, because without it the closed-contour value says nothing about the real integral. Only then are the residues worth computing, and the factor \( 2\pi i \) applied at the end
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