Question #41863

This problem orchestrates a 'Lethal Collision' between greatest integer functions and modulus-dependent logarithmic integration.

The primary trap lies in the argument of the logarithm, $2[x] - 2x + 1$, which simplifies to $1 - 2\{x\}$.

Students who fail to respect the modulus sign dependency will incorrectly evaluate the improper integral $\int_0^1 \ln(1 - 2t) dt$, often encountering undefined real values for $t > 1/2$ and abandoning the problem.

The modulus forces a split at $t = 1/2$, revealing an absolute value symmetry $\int_{-1}^1 \ln|u| du$.

Furthermore, the secondary trap involves the linear term $(2x - 1)$, which must be expanded as $(2k + 2t - 1)$.

Students frequently miss the odd symmetry of $(2t - 1)\ln|1 - 2t|$, leading to catastrophic integration by parts errors.

Finally, the summation bounds from $k=0$ to $89$ (not $90$) test boundary precision.