Fourier sine transform

The Fourier sine transform of f(x) is F_s(ω) = ∫₀^∞ f(x) sin ωx dx, with inverse f(x) = (2/π)∫₀^∞ F_s(ω) sin ωx dω.

Transform of e^(−x):

F_s(ω) = ∫₀^∞ e^(−x) sin ωx dx = [ e^(−x)(−sin ωx − ω cos ωx)/(1 + ω²) ]₀^∞
       = ω / (1 + ω²)

(In the symmetric convention F_s = √(2/π)·ω/(1+ω²); the result below is the same.)

Parseval's identity for sine transforms (with the convention above):

∫₀^∞ [f(x)]² dx = (2/π) ∫₀^∞ [F_s(ω)]² dω

Left side :  ∫₀^∞ e^(−2x) dx = 1/2
Right side:  (2/π) ∫₀^∞ ω² / (1 + ω²)² dω

Equating: (2/π) ∫₀^∞ ω²/(1 + ω²)² dω = 1/2, so ∫₀^∞ ω²/(1 + ω²)² dω = π/4.

Renaming the dummy variable ω as x:

Answer: ∫₀^∞ x² dx / (x² + 1)² = π/4.