We have Fourier sine transformation of f(x), such that,
F[f(x)] = [0,∞] ∫ f(x) sin(sx) dx = F(s)
Now, for f(x)= 1/x
F[f(x)] = [0,∞] ∫ (1/x) dx sin(sx) dx
Let
sx = y ⇒ x = y/s ⇒ dx = dy/s
Now,
F[f(x)] = [0,∞] ∫ (siny/y) dy = π/2
You can use fourier sine transform calculator to get instant and accurate results online.