Let $ \chi (w) = \frac{\pi}{j} 4\delta (w - 6) - \frac{\pi}{j} 4\delta (w + 6) $
Then $ x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \chi (w) e^{jwt}dw $
$ x(t) = \frac{1}{2\pi} [\frac{4\pi}{j}\int^{\infty}_{-\infty} \delta(w-6)e^{jwt} dw - \frac{4\pi}{j} \int_{-\infty}^{\infty} \delta(w+6)e^{jwt} dw] $
$ x(t) = \frac{2}{j}e^{j6t} - \frac{2}{j}e^{-j6t} = 4[\frac{e^{j6t} - e^{-j6t}}{2j}] = 4sin(6t) $
