Amazingly, the integral below has a simple solution:
Even more amazingly, the integral ultimately does not depend on the parameter . For several hours, I tried to figure out a way to demonstrate that is independent of , but I couldn’t figure out a way to do this without substantially simplifying the integral, but I’ve been unable to do so (at least so far).
So here’s what I have been able to develop to prove that is independent of without directly computing the integral .
I now multiply the top and bottom of this last integral by :
I now employ the substitution , so that . Since , the endpoints of integration do not change, and so
This final integral is independent of .