I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

Find the even Fourier transform.

I calculated the transform with this formula

Knowing that the function is even I write:

A bit integration gives, don't hesitate to call out errors.

Now as an application to this calculation I'm supposed to determine this integral.

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get