Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Analysis / 6 / iii

Set $w=i$, which satisfies $\Re w=0$, and take real parts of the absolutely convergent integral. Since $x^i=\cos (\log x)+i\sin (\log x)$ and $\cos (i\pi /2)=\cosh (\pi /2)$,Absolute integrability justifies moving the real-part operation through the integral.