Laplace Transform of Square Wave Function 
I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem that $$F(s) = \frac{1}{1-e^{-ps}} \int_0^p e^{-st}f(t)dt$$ where $f(t)$ is a periodic function with period $p$. From the graph and the information in the theorem, I deduce that the Laplace transform of the function can be calculated as follows: $$F(s) = \frac{1}{1-e^{-2as}} \int_0^{2a} e^{-st}f(t)dt = \frac{1}{1-e^{-2as}} \int_0^a e^{-st}dt\qquad (1)$$ because $f(t) = 1$ for $0 \le t \le a$, and $f(t) = 0$ for $a \le t \le 2a$. However, the book gets $$F(s) = \frac{1}{s(1+e^{-as})}\qquad (2)$$ 
Could someone lend me a hand with this problem?
Numbers added for convenience, thank you in advance mates. Fair winds.

$$
\frac{1}{1-e^{-2as}} \int_0^a e^{-st}dt=\frac{1-e^{-as}}{s(1-e^{-2as})}=\frac{1}{s(1+e^{-as})}
$$
