Solution
= Solution
The <function> is nonnegative and measurable, so <Lebesgue integrability> is equivalent to finiteness of its integral. For $\alpha\ne-1$,
$$
\int_\varepsilon^1x^\alpha\,dx=\frac{1-\varepsilon^{\alpha+1}}{\alpha+1}.
$$
Letting $\varepsilon\downarrow0$ gives a finite limit exactly when $\alpha>-1$; at $\alpha=-1$ the integral is $-\log\varepsilon$ and diverges. By the <monotone convergence theorem> for truncations,
$$
\boxed{\alpha>-1}.
$$