= Solution
For real <eigenfunctions> $y_i,y_j$, multiply their <Sturm-Liouville problems> crosswise and subtract. Integration gives
$$
(\lambda_i-\lambda_j)\int_1^R x^2y_iy_j\,dx
=\left[x(x-1)(y_iy_j'-y_i'y_j)\right]_1^R.
$$
At $R$, both <eigenfunctions> vanish, while at $1$ their bounded values and bounded first <derivatives> multiply a factor tending to zero. The right side is zero. Distinct <eigenvalues> therefore give the <orthogonality of Sturm-Liouville eigenfunctions>
$$
\boxed{\int_1^R x^2y_i(x)y_j(x)\,dx=0\qquad(\lambda_i\ne\lambda_j).}
$$
For complex <eigenfunctions>, replace $y_i$ by $\overline{y_i}$, in accordance with the <weighted inner product>.
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