Project ForgeEngineering Mathematics
Calculus
GATE 2016
Expert
The value of the integral ∫₋∞^∞ [sin(x) / (x² + 2x + 2)] dx evaluated using contour integration and the residue theorem is
- -π sin(1)/e
- B-π cos(1)/e
- Csin(1)/e
- Dcos(1)/e
Explanation
x²+2x+2=(x+1)²+1 has a pole at z=-1+i in the upper half-plane. For ∫e^(ix)/(x²+2x+2)dx, the residue at z=-1+i is e^(iz)/(2z+2) evaluated there, giving e^(-1)(cos1-i sin1)/(2i). Multiplying by 2πi and taking the imaginary part (since sin x = Im e^(ix)) gives -π sin(1)/e.
Source: GATE ME 2016 Set-1 (2016)
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