I'm aware of the way to quantify the following integral in the following way, however I'm trying to find another way to express the given integral. Especially when the function $f(x)$ is not particularly nice: $$F(x)=\int_0^x \lfloor{t}\rfloor f(t)dt=\sum_{k=1}^{x-1}\int_k^{k+1}kf(t)dt$$
Does anyone know of another way to solve this? (I'm fairly certain that the sum of the right is correct, if not can someone let me know)