If $f$ is a polynomial of degree $n$ such that $\int_0^1 f(x)x^kdx=0$ for all $k=1,2,\dots,n$, then $\int_0^1 f^2(x)dx=(n+1)^2\left(\int_0^1 f(x)dx\right)^2.$
This implies that $\begin{vmatrix} 1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} & \cdots & \frac{1}{n+1} \\ \frac{1}{2}& \frac{1}{3} & \frac{1}{4}& \cdots & \frac{1}{n+2} \\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5} & \cdots & \frac{1}{n+3}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{n+1}& \frac{1}{n+2}& \frac{1}{n+3}& \cdots & \frac{1}{2n+1}\\ \end{vmatrix}=0.$
Does it have a more direct algebraic proof?