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 :
Consider by changing we have so . Also , the similar integral but which is not zero again :
By considering the difference of two integrals we have
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 .