= Solution
This subpart is also present in the original PDF but missing from the local TeX. Put $p(x)=1-x^2$ and $\lambda_n=n(n+1)$. Multiply the <Legendre differential equation> for $P_n$ by $P_m$, multiply the equation for $P_m$ by $P_n$, and subtract. The <product rule> gives
$$
\frac{d}{dx}\bigl[p(P_mP_n'-P_nP_m')\bigr]
+(\lambda_n-\lambda_m)P_nP_m=0.
$$
Integrating from $-1$ to $1$, the boundary term vanishes: $p(\pm1)=0$, and the <Legendre polynomials> and their <derivatives> are finite there. For distinct nonnegative <integers> $m,n$, $\lambda_m\ne\lambda_n$. Thus the <Orthogonality of Legendre polynomials> is
$$
\boxed{\int_{-1}^1P_m(x)P_n(x)\,dx=0\quad(m\ne n)}.
$$
The vanishing coefficient at the endpoints causes no difficulty here because the solutions are <polynomials>.
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