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
$$
\begin{equation*}
L_1(x) = x.
\end{equation*}
$$