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[tex]\mathcal{L} = \frac{m}{2} \dot{x}(t)^2[/tex]

In calculating it's classical action, they perform a simple integral over some interval of time [itex]t_a \rightarrow t_b[/itex].

[tex] S_{cl} = \frac{m}{2} \int_{t_b}^{t_a} \dot{x}(t)^2 \; dt = \frac{m}{2} \frac{(x_b - x_a)^2}{t_b - t_a}[/tex]

I don't see how that result follows! Is there some nifty integration by parts that I'm missing?