Correct answer is (b) πδ(ω) + \(\frac{1}{jω}\)
To explain: We know that sgn(t) = 2u(t) – 1.
u(t) = \(\frac{1}{2}\)[sgn(t)+1]
Its Fourier transform is F[u(t)] = \(\frac{1}{2}\) F[sgn(t)] + \(\frac{1}{2}\) F[1]
As the Fourier transforms F[1] = 2πδ(ω) and [sgn(t)] = \(\frac{2}{jω}\), hence
F[u(t)] = πδ(ω) + \(\frac{1}{jω}\).