Laplace Transforms · Challenge problem

A square wave has period $2$: $f(t) = 1$ for $0 \le t < 1$ and $f(t) = 0$ for $1 \le t < 2$, repeated. Using the periodic-function formula $\mathcal{L}\{f\} = \dfrac{1}{1 - e^{-2s}}\displaystyle\int_0^2 f(t)e^{-st}\,dt$, find $\mathcal{L}\{f\}$ at $s = \ln 2$.