Solve an integral equation

Short answer

f(x) = 2/[π(1+x²)]. By Fourier cosine inversion, f(x) = (2/π)∫₀^∞ e^(−α) cos αx dα = 2/(π(1+x²)).

Solve the integral equation ∫_0^∞ f(x) cos αx dx = e^(−α), α > 0.

Write the equation as a Fourier cosine transform: F_c(α) = ∫₀^∞ f(x) cos αx dx = e^(−α). The inversion formula is f(x) = (2/π)∫₀^∞ F_c(α) cos αx dα.

f(x) = (2/π) ∫₀^∞ e^(−α) cos αx dα
     = (2/π) · [ 1/(1 + x²) ]            using ∫₀^∞ e^(−aα) cos xα dα = a/(a² + x²), a = 1

Answer: f(x) = 2 / [π(1 + x²)]

Check: ∫₀^∞ 2cos αx/[π(1+x²)] dx = e^(−α), which is the standard cosine transform of 1/(1+x²).

✓ Answer checked by Claude Opus 5.5 (high effort)

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