Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S1 / 2 / 3

Taking the difference of the two bank integrals at $n=2$ givesThe $I_j$ are real and convergent. Comparing real and imaginary parts proves $\boxed{I_0=\pi /2,I_1=0}$. Independently, $x\mapsto 1/x$ changes the sign of each odd logarithmic moment and preserves the measure, so every odd $I_j$ vanishes.