Floors and Ceilings 2 - Room on Ceiling, Floor on Room

Calculus Level 5

\[ \large \int_0^\infty \left ( \frac x{\lceil x \rceil} - \frac {\lfloor x \rfloor}x \right) \, dx \]

Given that the integral above can be expressed as \(\frac pq \ln(q \pi) - \frac rs \gamma ^t \) for natural numbers \(p,q,r,s,t\) with \(r,s\) coprime.

Find the value of the \(p+2q+3r+4s+5t\).

Details and Assumptions:

\(\gamma\) is the Euler-Mascheroni constant, which is equals to \( \displaystyle \lim_{m\to\infty} \left(-\ln(m+1) + \sum_{n=1}^m \frac1n \right) \).


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