I'm solving the problem $$-\frac{d^2u}{dx^2} + w^2 u = e^{-|x|}$$ subject to $u(x) \rightarrow 0$ as $|x| \rightarrow \infty$ and $w \neq \pm 1, w \in \mathbb{R}$ a constant on $-\infty < x < \infty$ using Fourier transform methods.
Taking a Fourier transform gives $k^2 \tilde{u}(k) + w^2 \tilde{u}(k) = \frac{2}{1 + k^2}$ so
$$\tilde{u}(k) = \frac{2}{(1+k^2)(w^2+k^2)}$$
It remains to invert the Fourier transform. I am unsure how to do this since $w$ is not restricted to be positive, so I cannot invert $\frac{1}{w^2+k^2}$ as something like $\frac{1}{2w}e^{-w|x|}$.
How can this Fourier transform be inverted?