We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function:
$$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$
is integrable on the interval $]-1,1[$.