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 $$\begin{equation} L_2(x) = \frac{1}{2}\left(3x^2-1\right). \tag{12} \end{equation}$$

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