Fourier Transform

Find the Fourier cosine transform of f(x) = e^−x, x > 0, and hence show that ∫₀^∞ 1(1 + x²)² dx = π4.

Find the Fourier transform of the function e^−x2, and also verify the convolution theorem for the functions f(x) = e^−x2 and g(x) = e^−x2.

Find the Fourier sine transform of f(x) = e^−x, x ≥ 0, and hence evaluate ∫₀^∞ x sin x1 + x² dx.