Extension: What Is Interpolation?

Given a function $f(x)$, the function $x\mapsto \int_0^x f(t)\,dt$ is an anti-derivative of $f(x)$ vanishing at $x=0$. Continuing the process, the second anti-derivative (vanishing at $x=0$) is $x\mapsto \int_0^x f(t)(x-t)\,dt$, the third anti-derivative (vanishing at $x=0$) is $x\mapsto \int_0^x f(t)\frac{(x-t)^2}{2!}\,dt$, and so on. Now, here's a question about these "repeated anti-derivatives".