Integral to Infinity of Sine p x over x/Proof 2

Theorem

$\ds \int_0^\infty \frac {\sin p x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > 0 \\ 0 & : p = 0 \\ -\dfrac \pi 2 & : p < 0 \end {cases}$

Proof

This theorem requires a proof. In particular: Use Primitive of Sine of a x over x, and also the analytic solution as found in Integration, 2nd ed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.