Sequence of integrals defined by recurrence

Question

For a sequence of integrals defined as follows

$F_0(x)=f(x)$ for some function $f(x)$, $F_n(x)=\int_0^x F_{n-1}(y)dy$ for all $n\geq1$,

can we use change of variables to find a nice expression for $F_2(x)$ that only includes 1 integration?

I've tried the following:

$F_2(x)=\int_0^x F_1(y)dy=\int_0^x \int_0^y f(x)dxdy$ and drawing a graph of the region I'm integrating on but I'm doing nonsense... Can you help me? I can feel it's possible to simplify the expression but how?

Thank you!

Answer 1

You could try integration by parts to simplify the integral. $$F_2(x) = \int_{0}^x F_1(y) dy = F_1(x) \cdot x - \int_0^x F_1'(y) y dy = xF_1(x) - \int_0^x yf(y)dy$$

Then $$F_1(x) = \int_0^x f(y)dy$$ so we have $$F_2(x) = \int_0^x (x - y) f(y) dy$$