Past exam of the options II course of the University of Oxford / 2009 / D / D3 / i

The function is nonnegative and measurable, so Lebesgue integrability is equivalent to finiteness of its integral. For $\alpha \ne -1$,Letting $\epsilon \downarrow 0$ gives a finite limit exactly when $\alpha >-1$; at $\alpha =-1$ the integral is $-\log \epsilon$ and diverges. By the monotone convergence theorem for truncations,

For $\alpha \ne -1$,The limit as $R\to \infty$ is finite exactly when $\alpha <-1$; the case $\alpha =-1$ diverges logarithmically. Positivity and the monotone convergence theorem identify these limits with the Lebesgue integral. Hence