First off, I'd like to say that I love your videos!
The reason I am writing: I ran across Feynman's "differentiation under the integral" trick years ago when I went on a bit of a Feynman kick (recommend! a brilliant man...). It recently popped up again on my fb here:
If you haven't already, his technique might be worth a look, and perhaps a video? Just a thought. Anyway, keep on doing good things! Sincerely, James