A bit of integration involving a continuous function

A bit of integration involving a continuous function

If $f$ is continuous on $[0,1]$ and if $\int_{0}^{1}f(x)x^ndx=0$ for
$n=0,1,2,3...$then $\int_{0}^{1}f^2(x)dx=0$.
This is how I proceeded. For $n=1$, $\int_{0}^{1}f(x)xdx=0$ . (Using by
parts) $\implies$ $x\int f(x)dx]_0^1-\int_{0}^{1}(\int f(x)dx)dx=0$. Let
$I(x)=\int f(x)dx$. Then $I(1)=\int_{0}^{1}I(x)dx$. For $n=2$ we have
$\int_{0}^{1}f(x)x^2dx= x\int xf(x)dx]_0^1-\int_{0}^{1}(\int xf(x)dx)dx=0$
In general for any $n$, $\int_{0}^{1}f(x)x^ndx=x\int
x^{n-1}f(x)dx]_0^1-\int_{0}^{1}(\int x^{n-1}f(x)dx)dx=0$.
Then I don't see how I can use this information to get the desired result.