Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / D / 2 / a

There is no singularity at the lower endpoint one. Put $t=\log (1+x)$, so $dx=e^{t}dt$. The nonnegative integral becomesFor $\alpha =1$, it converges as a reciprocal-square integral. For $\alpha >1$, exponential decay makes it converge. For $\alpha <1$, its integrand eventually grows without bound, so its integral diverges. ThusThis is the logarithmic borderline in a power-tail integrability test.