Sine transform of e^{-x}; ∫x²/(1+x²)²

Definition. F_s(s) = √(2/π) ∫₀^∞ f(x) sin sx dx.

Sine transform of e⁻ˣ

Using ∫₀^∞ e^{−ax} sin sx dx = s/(a² + s²) with a = 1:

F_s(s) = √(2/π) · s/(1 + s²)

Using Parseval's identity

For sine transforms, ∫₀^∞ [f(x)]² dx = ∫₀^∞ [F_s(s)]² ds.

• Left side: ∫₀^∞ e^{−2x} dx = ½
• Right side: (2/π) ∫₀^∞ s²/(1 + s²)² ds

(2/π) ∫₀^∞ s²/(1 + s²)² ds = 1/2
∫₀^∞ s²/(1 + s²)² ds       = π/4

Renaming the variable s to x:

∫₀^∞ x²/(1 + x²)² dx = π/4 ∎

(Numerical check: 0.785398 ✓. This also agrees with ∫₀^∞ dx/(1 + x²) − ∫₀^∞ dx/(1 + x²)² = π/2 − π/4.)