= Solution
By <parity preservation by polynomial least-squares projection>, $p_2(x)=\beta x$: the constant and quadratic coefficients vanish. The remaining <normal equation> is
$$
0=\int_{-1}^1(x^{1969}-\beta x)x\,dx=\frac2{1971}-\beta\frac23.
$$
Therefore
$$
\boxed{p_2(x)=\frac3{1971}x=\frac{x}{657}.}
$$
The residual is automatically <orthogonal> to $1$ and $x^2$ by parity and to $x$ by construction, so it is orthogonal to every element of $\Pi_2$. This checks the full <linear least squares> condition, rather than only a necessary coefficient equation.
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