We're doing fourier integrals in class, but unfortunately I have no idea how to even begin to tackle this one. The examples we have done in class were way simpler than this one:
$$ \int_0^\infty \dfrac {w^3\sin(xw)}{w^4+4}dw = \dfrac{1}{2}\pi e^{-x}\cos(x) $$ when x > 0
Any ideas?
I would take its Fourier sine transform of $e^{-x}\cos x$ and see that it's consistent with what's on the left. $$ \int_0^{\infty} e^{-x}\cos x \sin w x\,dx =\int_0^{\infty} e^{-x}\frac{e^{ix}+e^{-ix}}{2}\frac{e^{iwx}-e^{-iwx}}{2i}\,dx$$ Multiply out exponentials and evaluate: $$\frac{-1}{4i}\left(\frac{1}{-1+i+iw}-\frac{1}{-1+i-iw}+\frac{1}{-1-i+iw}-\frac{1}{-1-i-iw}\right)$$ Which collects into $$\frac{-1}{4i}\left(\frac{-2iw}{(-1+i)^2+ w^2}+\frac{-2iw}{(-1-i)^2+ w^2} \right)$$ simplifies to $$\frac{w}{2}\left(\frac{1}{-2i + w^2}+\frac{1}{2i+ w^2} \right)$$ and boils down to $$\frac{w}{2} \frac{2w^2}{4 + w^4} $$