$\int f(x) x^n dx=0$ for all $n\geq 1$.

Question

This problem is similar to Baby Rudin Excercise 20 in chapter 7. The problem states that $g:[0,1]\to \mathbb{R}$ is continuous and that for all natural $n\geq 1$, $\int_0^1 g(x) x^n dx=0$. Prove that $g\equiv 0$. We do NOT know that $\int_0^1 g(x)dx=0$. (The Rudin exercise has the additional assumption that this holds for $n=0$ as well) I know that this implies that $\int_0^1 g(x)dx=\int_0^1 g(x)^2dx$ but am unable to conclude from here.

Answer 1

Hint. We know that $\int_0^1 (xg(x)) x^n dx=0$ for all $n\geq 0$. Now apply Baby Rudin Exercise to the continuous function $xg(x)$.