$\int_0^n s (\int_0^s f(t) \, dt) \, ds = \int_0^n f(s) (\int_s^n t \, dt) \, ds $
The $s$ and $t$ in $f$ are just dummy variables of the same function.
Is there a way to start from the left hand side of this equation and show it is equal to the right hand side or vice-versa? The author of the paper that uses this relation only mentions the use of Fubini's Theorem to show it.
There is a geometric interpretation of this problem here with $f = 1$: Rearranging double integral and bounds
Edit: I see the process of it shown here without $s$ in the outer integral: Double integral and variable change problem