Inspired by Abhishek Singh's Integral!

Calculus Level 4

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

If the above integral is of the form aπbc\frac{a\pi^{b}}{c} where aa and cc are coprime and a,b,cZ+a,b,c \in \mathbb{Z}^{+}, find a+b+ca+b+c.

Details and Assumptions

{x}\lbrace x \rbrace denotes the fractional part of xx such that {x}=xx\lbrace x \rbrace = x-\lfloor x \rfloor.

You might want to try this problem first.


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