Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / D / 3 / c / i / Solution

Substitute $u=\log x$, giving $dx=e^{u}du$. The nonnegative integral becomesFor $\alpha >1$ an integration by parts gives $1/(\alpha -1{)}^2$; its boundary term vanishes because exponential decay dominates $u$. Hence it is finite, proving integrability and