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The value of the integral $$\displaystyle{\int_{-\infty }^{\infty }\frac{\sin x}{x^2+2x+2}}dx$$ evaluated using contour integration and the residue theorem is

  1. $\displaystyle{\frac{-\pi \sin(1)}{e}}\\$
  2. $\displaystyle{\frac{-\pi \cos (1)}{e}} \\$
  3. $\displaystyle{\frac{\sin (1)}{e}} \\$
  4. $\displaystyle{\frac{\cos (1)}{e}}$
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