Dirichlet Integral/Proof 1

Proof
By Fubini's Theorem:


 * $\displaystyle \int_0^\infty \left({\int_0^\infty e^{- x y} \sin x \rd y}\right) \rd x = \int_0^\infty \left({\int_0^\infty e^{- x y} \sin x \rd x}\right) \rd y$

Then:

and:

Hence: