Past exam of the options II course of the University of Oxford / 2008 printed variant / D / D3 / c

Substitute $t=\log x$, with $dx=e^{t}dt$. The nonnegative integral becomesby scaling and integration by parts. The exponential decays because $\alpha >1$, with zero boundary term at infinity, proving integrability and

For $x>1$, the geometric series givesAll terms are nonnegative, so monotone convergence permits termwise integration. Part (i), with $\alpha =p+k+1>1$, givesThis series is finite for $p>0$, which also proves integrability of the original nonnegative function.