Solution
= Solution
The function is nonnegative and measurable. For $\alpha\ne-1$, integrate from $\varepsilon$ to one to get $(1-\varepsilon^{\alpha+1})/(\alpha+1)$. The limit is finite exactly when $\alpha+1>0$. At $\alpha=-1$ the integral is $-\log\varepsilon\to\infty$. Thus $\boxed{x^\alpha\in L^1(0,1)\iff\alpha>-1}$.