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