Solution
= Solution
For $x>1$, the geometric series gives
$$
\frac{\log x}{x^p(x-1)}=\sum_{k=0}^\infty x^{-(p+k+1)}\log x.
$$
All terms are nonnegative, so monotone convergence permits termwise integration. Part (i), with $\alpha=p+k+1>1$, gives
$$
\boxed{\int_1^\infty\frac{\log x}{x^p(x-1)}\,dx=\sum_{k=0}^\infty\frac1{(p+k)^2}}.
$$
This series is finite for $p>0$, which also proves integrability of the original nonnegative <function>.