Computing the Inverse Fourier Transform
$ \ X(\omega)= 8 \pi w \delta(w-9) + 2 \pi w^{3} \delta(w-4 \pi) $
The inverse Fourier transform is defined as:
$ x(t) = \int_{-\infty}^{\infty} \frac{X(w)}{2 \pi} e^{jwt} dw $
Using this formula to determine the signal:
$ \ x(t) = \frac{8 \pi}{2 \pi} \int_{-\infty}^{\infty} w e^{jwt} \delta(w-9) dw + \frac{2}{2 \pi} \int_{-\infty}^{\infty}w^{3} \delta(w-4 \pi) e^{jwt} dw $
Now using the sifting property of the delta function we find that the signal is
$ \ x(t) = 36 e^{j9t} + 64 \pi^{2} e^{j4\pi t} $
