Solution
= Solution
For the weight $w=x$, substitution $x=e^s$ gives
$$
\langle y_m,y_n\rangle_w=\int_1^e\frac{\sin(m\pi\log x)\sin(n\pi\log x)}x\,dx=\int_0^1\sin(m\pi s)\sin(n\pi s)\,ds.
$$
The <product-to-sum formula> makes this zero when $m\ne n$ and $1/2$ when $m=n$. Therefore
$$
\boxed{\langle y_m,y_n\rangle_w=\tfrac12\delta_{mn}.}
$$
In particular the <eigenfunctions> are <orthogonal>, and multiplying each by $\sqrt2$ gives an <orthonormal basis> for the weighted space.