This is related to the integrated tail probability expectation formula:
$$ X= \int_0^X dx = \int_0^\infty 1_{X>x} dx,$$
followed by
$$ E[X] = E \left[ \int_0^\infty 1_{X>x} dx \right] = \int_0^\infty E[1_{X>x}] dx $$ $$=\int_0^\infty P(X>x) dx = \int_0^\infty \left( \int_x^\infty p(z) dz\right) dx$$
Similarly, for deterministic $f$, we have:
$$ \int_0^X f(x) dx = \int_0^\infty 1_{X>x} f(x) dx,$$
followed by
$$ E \left[ \int_0^X f(x) dx \right] = E \left[ \int_0^\infty 1_{X > x} f(x) dx \right] = \int_0^\infty E[1_{X> x} f(x)] dx $$ $$ =\int_0^\infty P(X> x) f(x) dx =
\int_0^\infty \left( \int_x^\infty p(z) dz\right) f(x) dx $$