$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Question

Let $t,x$ be nonnegative reals.

Let $* ^{[k]}$ denote k th iteration.

Find real-analytic $f(x)$ such that

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Holds.

We require analytic iterations. ( $ f^{[t]}(0) $ is analytic in $t$ )

By analogue also

$\sum_{x=0}^t g(x) - x = g^{[t]}(0) - t - 1$

Find Such $f(x),g(x)$.