f(x)=\frac{1}{\pi}\int_{0}^{+\infty}du\int_{-\infty}^{+\infty}f(\xi)\cos(u(x-\xi))d\xi=
\frac{1}{\pi}\int_{-\infty}^{+\infty}\int_0^{+\infty}f(\xi)\cos(u(x-\xi))\,du\,d\xi=
\frac{1}{\pi}\lim_{T\rightarrow+\infty}
\int_{-\infty}^{+\infty}\int_0^{T}f(\xi)\cos(u(x-\xi))\,du\,d\xi=
\frac{1}{\pi}\lim_{T\rightarrow+\infty}
\int_{-\infty}^{+\infty}\frac{\sin(T(x-\xi))}{x-\xi}f(\xi)\,d\xi.