Use inverse Fourier transform to find the Green's function of the 1D wave equation?

Question

I need to find the Green's function $g$ of the 1D wave equation $$g_{tt} = g_{xx}, \quad g(x, 0) = 0, g_t(x, 0) = \delta(x).$$ I tried doing the Fourier transform w.r.t. $x$ on the equation to get $$ G''(t) = \color{red}{-}k^2 G(t), \quad G(0) = 0, G'(0) = 1/\sqrt{2\pi}.$$ Solving the 2nd order ODE of $G$ should give $$G(t) = -i(e^{ikt} - e^{-ikt})/(2k\sqrt{2\pi}).$$ But I'm having trouble doing the inverse Fourier transform: $$g(x, t) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \dfrac{e^{ikx} (-i) (e^{ikt} - e^{-ikt})}{2k\sqrt{2\pi}} \,dk = \dfrac{1}{4\pi}\int_{-\infty}^{\infty} \dfrac{(-i)(e^{ik(x+t)} - e^{ik(x-t)})}{k} \,dk = \cdots?$$ The answer I got from here should be $$(\rho(x + t) + \rho(x - t))/4, \quad \rho(> 0) = 1, \rho(\leq 0) = 0$$ that $\rho(\cdot)$ indicates whether the input is positive. Am I solving the ODE wrong or putting the wrong stuff into the integral? Thanks in advance.