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The value of the integral ∫ from −∞ to ∞ of [sin x / (x² + 2x + 2)] dx evaluated using contour integration and the residue theorem is

1. −π sin(1)/e
2. −π cos(1)/e
3. sin(1)/e
4. cos(1)/e

Correct answer: −π sin(1)/e

Solution

The integral evaluates to −π sin(1)/e because the integrand has poles that contribute to the residue calculation, and the sine function in the numerator leads to a negative sign in the final result when applying the residue theorem.

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