We have $I(\alpha)=\int_{0}^{1} \frac{x-1}{x^\alpha-1}dx$, and the problem asks to study convergence of $I$ according to values of $\alpha$.
2026-03-30 14:39:04.1774881544
Study convergence of integral $\alpha$ according to parameter $\alpha$
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It is not defined for $\alpha =0$ and it is convergent for all $\alpha \neq 0$. For justification look at the limiting values of the integrand as $x \to 0$ and $x \to 1$. [Use L'Hopital's Rule].