IF you have a density, then
So, it's correct. Next use dF(x) instead of f(x) and switch the order of integration again...
The second one looks like parts.
I have a problem, which I assume is easy to solve, but I don't know exact how to write a coherent argument...
1. For any distribution function and any , we have .
2. For any compactly supported function and any probability measure with cumulative function F, one has .