Solution
= Solution
Taking the difference of the two bank <Integral>[integrals] at $n=2$ gives
$$
\int_0^\infty\frac{(\log x)^2-(\log x+2\pi i)^2}{1+x^2}\,dx
=4\pi^2 I_0-4\pi i I_1=2\pi^3.
$$
The $I_j$ are real and convergent. Comparing real and <Imaginary part>[imaginary parts] proves $\boxed{I_0=\pi/2,\ I_1=0}$. Independently, $x\mapsto1/x$ changes the sign of each odd logarithmic moment and preserves the measure, so every odd $I_j$ vanishes.