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

A. $$\frac{{ - \pi \sin \left( 1 \right)}}{{\text{e}}}$$

B. $$\frac{{ - \pi \cos \left( 1 \right)}}{{\text{e}}}$$

C. $$\frac{{\sin \left( 1 \right)}}{{\text{e}}}$$

D. $$\frac{{\cos \left( 1 \right)}}{{\text{e}}}$$

Answer: Option A