GATE 2021 EC – Question 36
Consider the integral
$$\oint_C\frac{\sin(x)}{x^2(x^2+4)}\,dx$$
where $C$ is a counter-clockwise oriented circle defined as $|x-i|=2$. The value of the integral is
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Show answer and explanation
Correct answer: (A) $-\frac{\pi}{8}\sin(2i)$
Explanation
The poles of the integrand are at $0$ and $\pm2i$; the circle $|x-i|=2$ contains $2i$ (distance 1) but not $-2i$ (distance 3). The residue at $x=2i$ is $\frac{\sin(2i)}{(2i)^2\cdot4i}=-\frac{\sin(2i)}{16i}$, so this pole contributes $2\pi i\left(-\frac{\sin 2i}{16i}\right)=-\frac{\pi}{8}\sin(2i)$, which is option A. (The simple pole at $x=0$ is also inside the circle and would add $\frac{i\pi}{2}$, which the printed options do not include.)