Dirichlet Integral/Proof 4

Proof
From Integral to Infinity of Function over Argument:


 * $\ds \int_0^\infty {\dfrac {\map f x} x} = \int_0^{\to \infty} \map F u \rd u$

for a real function $f$ and its Laplace transform $\laptrans f = F$, provided they exist.

Let $\map f x := \sin x$.

Then from Laplace Transform of Sine:
 * $\laptrans {\map f x} = \dfrac 1 {s^2 + 1}$

Hence: