Using Fourier cosine/sine integral , compute an integration

Question

For all $x>0$, show that

$$\int_0^{\infty} \frac{w^3 \cos(w x)}{w^4 +4} dw = \frac{\pi}{2} e^{-x}\sin(x)$$

I want to solve this problem using Fourier integral, but it is too hard for me.