Orthogonal polynomials, Legendre

For $L_1(x)$ we have the general expression $$\begin{equation*} L_1(x) = a+bx, \end{equation*}$$ and using the orthogonality relation $$\begin{equation*} \int_{-1}^1L_0(x)L_1(x)dx=0, \end{equation*}$$ we obtain $a=0$ and with the condition $L_1(1)=1$, we obtain $b=1$, yielding $$\begin{equation*} L_1(x) = x. \end{equation*}$$

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