Sine transform of e^{-x}; ∫x sin mx/(x²+1)

Definition. F_s(s) = √(2/π) ∫₀^∞ f(x) sin sx dx, and the inversion is f(x) = √(2/π) ∫₀^∞ F_s(s) sin sx ds.

Sine transform of e⁻ˣ

∫₀^∞ e^{−x} sin sx dx = s/(1 + s²), so

F_s(s) = √(2/π) · s/(1 + s²)

Deducing the integral

By the inversion formula, for x > 0:

e^{−x} = √(2/π) ∫₀^∞ [ √(2/π) s/(1 + s²) ] sin sx ds
       = (2/π) ∫₀^∞ s sin sx /(1 + s²) ds

So ∫₀^∞ s sin sx /(1 + s²) ds = (π/2) e^{−x}. Rename the variables: s → x and x → m (m > 0):

∫₀^∞ x sin mx /(x² + 1) dx = (π/2) e^{−m} ∎

The case m = 1

∫₀^∞ x sin x /(1 + x²) dx = π/(2e) ≈ 0.5779