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GATE 2016 ME (ME1) – Question 38

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

The value of $\int_{-\infty}^{\infty}\sin x/(x^2+2x+2)\,dx$, evaluated by contour integration and residues, is ______.

  1. $-\pi\sin(1)/e$
  2. $-\pi\cos(1)/e$
  3. $\sin(1)/e$
  4. $\cos(1)/e$

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Show answer and explanation

Correct answer: (A) $-\pi\sin(1)/e$

Explanation

Use the imaginary part of the integral of exp(iz)/[(z+1)²+1], closing the contour in the upper half-plane. The enclosed pole is z=−1+i, with residue exp(i(−1+i))/(2i). Thus the complex integral isπexp(−1−i), whose imaginary part is **−πsin1/e, A**. The exponential decays on the chosen semicircle.