Question 30
Find all non constant polynomial functions $f: [0, 1] \to \mathbb{R}^*$, having rational coefficients, for which the following property holds: for every $x$ in $[0, 1]$, there exist two polynomial functions $g_x, h_x: [0, 1] \to \mathbb{R}$, with rational coefficients, such that $h_x(x) \neq 0$ and
$$
\int_0^x \frac{1}{f(t)} dt = \frac{g_x(x)}{h_x(x)}.
$$
$$
\int_0^x \frac{1}{f(t)} dt = \frac{g_x(x)}{h_x(x)}.
$$