Inspired by Abhishek Singh's Integral!

Calculus Level 4

\[\int_{0}^{\infty} \frac{\lbrace x \rbrace^{\lfloor x \rfloor}}{\lceil x \rceil} \ \mathrm{d}x\]

If the above integral is of the form \(\frac{a\pi^{b}}{c}\) where \(a\) and \(c\) are coprime and \(a,b,c \in \mathbb{Z}^{+}\), find \(a+b+c\).

Details and Assumptions

\(\lbrace x \rbrace\) denotes the fractional part of \(x\) such that \(\lbrace x \rbrace = x-\lfloor x \rfloor\).

You might want to try this problem first.


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