Elementary solution to an integral that is equal to Pi over 2E

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{Integrate[xExp[I x]/(1 + x^2), {x, 0, Infinity}] // Im,
InverseMellinTransform[
Integrate[
MellinTransform[xSin[a x]/(1 + x^2), a, s], {x, 0, Infinity},
Assumptions → 0 < Re[s] < 2], s, a] /. a → 1,
InverseMellinTransform[
Integrate[
MellinTransform[xSin[x]/(a + x^2), a, s] // PowerExpand, {x, 0,
Infinity}, Assumptions → -(1/2) < Re[s] < 1/2], s, a] /. a → 1,
InverseLaplaceTransform[
Integrate[
LaplaceTransform[xSin[a x]/(1 + x^2), a, s], {x, 0, Infinity},
Assumptions → Re[s] > 0], s, a] /. a → 1}

  (*{π/(2 E), π/(2 E), π/(2 E), π/(2 E)}*)
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