if $x > 0$, show that

$$f(x)=\int_0^\infty \frac{\cos(xt)}{1+t^2} \, dt= \frac{\pi}{2}e^{-x}$$

The problem appears in the book Differential Equations: With Applications and Historical Notes, under the chapter derivative and integral of Laplace transformation. So I tried to apply the equation

$$\int_0^\infty \frac {f(t)} t \, dt = \int_0^\infty F(p)\,dp,$$

where $F(p) = L[f(x)](p)$, the Laplace transformation of $f(t)$. However, I failed to manipulate the expression to apply this equality.

I need your help. Please not be misled by my approach, it might be totally wrong.