An integral with the floor function

Calculus Level 4

\[\large \int_1^\infty\frac{\lfloor x\rfloor}{x^3}\, dx = \frac{3}{a}+\frac{\pi b}{12}+\frac{\pi^2 c}{24}-1\]

The equation above holds true for where \(a,b\) and \(c\) are integers \(a\), \(b\) and \(c\). Compute the value of \(c^{a+b}\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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