Second order reaction cases
If you are talking about the 1.13 and 1.14 equations, than whta he did was just a simple subistituction where $[A] = x,\ d[A] = dx$, if not, please identify the step.

And, yes, you are going to get diferent equations for $[A]_{(t)}$ depending on the order of reaction, $0^{th}$ order gives you a linear equation, $1^{st}$ order gives you an exponetial equation, and all the orders higher than 1 gives you a fraction diference like: $$\frac{1}{A_t^n}-\frac{1}{A_0^n} = n\ k\ t$$ where n is the order of reaction minus 2.

The step where they make $[A]^2$ is only because you have an elementary equation, it can not be assumed to be that if the equation was golbal or not elementary.