I was calculating the so-called [Dirichlet Integral][1],

$$\displaystyle \int_{0}^{\infty}\frac{\sin x}{x} \text{d} x$$

and then I wondered about his name and history: 

1. Why it has the name of [Dirichlet][2]? 

2. When was it first calculated? 

3. In what context it showed up for the first time and from which interests it needed (if so) to be evaluated?

I don't know any resources (except for Stack Exchange) for researching this questions, and I haven't found anything related to them after surfing along the first 4 pages of results of "dirichlet integral history" in Google.

Does anybody know anything about its history or where to find information about it?

  [1]: https://en.wikipedia.org/wiki/Dirichlet_integral
  [2]: https://es.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet