Editing Dirichlet integral (section)

=== Laplace transform ===
Let <math>f(t)</math> be a function defined whenever <math>t \geq 0.</math> Then its [[Laplace transform]] is given by
<math display="block">\mathcal{L} \{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \,dt,</math>
if the integral exists.<ref>{{Cite book |last=Zill|first=Dennis G. |title=Differential Equations with Boundary-Value Problems |url=https://archive.org/details/differentialequa00zill_769|url-access=limited|last2=Wright|first2=Warren S. |publisher=Cengage Learning |year=2013 |isbn=978-1-111-82706-9|pages=[https://archive.org/details/differentialequa00zill_769/page/n323 274]-5 |chapter=Chapter 7: The Laplace Transform}}</ref> 

A property of the [[Laplace transform#Evaluating improper integrals|Laplace transform useful for evaluating improper integrals]] is
<math display="block"> \mathcal{L} \left [  \frac{f(t)}{t} \right] = \int_{s}^{\infty} F(u) \, du,
</math>
provided <math>\lim_{t \to 0} \frac{f(t)}{t}</math> exists.

In what follows, one needs the result  <math>\mathcal{L}\{\sin t\} = \frac{1}{s^2 + 1},</math> which is the Laplace transform of the function <math>\sin t</math> (see the section 'Differentiating under the integral sign' for a derivation) as well as a version of [[Abel's theorem]] (a consequence of the [[Final value theorem#Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)|final value theorem for the Laplace transform]]).

Therefore,
<math display="block"> 
\begin{align}
\int_{0}^{\infty} \frac{\sin t}{t} \, dt
&= \lim_{s \to 0} \int_{0}^{\infty} e^{-st} \frac{\sin t}{t} \, dt
= \lim_{s \to 0} \mathcal{L} \left [  \frac{\sin t}{t} \right] \\[6pt]
&= \lim_{s \to 0} \int_{s}^{\infty} \frac{du}{u^2 + 1}
= \lim_{s \to 0} \arctan u \Biggr|_{s}^{\infty} \\[6pt]
&= \lim_{s \to 0} \left[ \frac{\pi}{2} - \arctan (s)\right]
= \frac{\pi}{2}.
\end{align} </math>