Orthogonal polynomials, Legendre
We can proceed in a similar fashion in order to determine
the coefficients of \( L_2 \)
$$
\begin{equation*}
L_2(x) = a+bx+cx^2,
\end{equation*}
$$
using the orthogonality relations
$$
\begin{equation*}
\int_{-1}^1L_0(x)L_2(x)dx=0,
\end{equation*}
$$
and
$$
\begin{equation*}
\int_{-1}^1L_1(x)L_2(x)dx=0,
\end{equation*}
$$
and the condition
\( L_2(1)=1 \) we would get
$$
\begin{equation}
L_2(x) = \frac{1}{2}\left(3x^2-1\right).
\tag{12}
\end{equation}
$$