Perhaps I am driving on the detour , you will see that I am actually making use of the calculus of the parameter , together with Laplace Transform .

Let's consider , first we make a substitution then we have

at this moment i remember that the Laplace Transform of is by changing the dummy variables we have :

Following it is the change of the order of integration :

Sub.

Consider by changing we have so . Also , the similar integral but which is not zero again :

, sub

By considering the difference of two integrals we have

or

Finally ,

Remarks :

If we consider the sum of these two integrals , we obtain

this result helps evaluate the integral which can be started in a quite similar way .