Unusual Laplace Transform

Calculus Level 5

Let \(g(s)\) be the Laplace Transform of the function \(f(x) = \dfrac{1}{\lceil x \rceil}\).

What is the closed form of \(g\) using the variable \(s\)?

\[\begin{array}{ccc} A: & \dfrac{(1-e^s)\ln(1-e^{-s})}{s} \\ B : & \dfrac{\ln(e^s-1)}{s^2} \\ C : & \dfrac{s \ln(e^s-1)}{e^s-1} \\ D : & \dfrac{\ln(1-e^{-s})}{s(e^{-s}-1)} \end{array}\]

Notation: \( \lceil \cdot \rceil \) denotes the ceiling function.

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