This is a straightforward application of the steepest descent method. So, we have the integral (taken from the lectures you cite)

$$Q_N=\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^\infty d\mu e^{[Nq(\beta J,\beta b,\mu)]}$$

being

$$q(\beta J,\beta b,\mu)=\ln\{2\cosh[\beta(J\mu+b)]\}-\frac{\beta J\mu^2}{2}.$$

Now, you will have, by applying steepest descent method to the integral,

$$\frac{\partial q}{\partial\mu}=\beta J\tanh[\beta(J\mu+b)]-\beta J\mu=0.$$

Let us call $\mu_s$ the solution of this equation and expand the argument of the exponential around this value. You will get

$$q(\beta J,\beta b,\mu)=q(\beta J,\beta b,\mu_s)-\frac{J\beta}{2}[1-J\beta(1-\mu_s^2)](\mu-\mu_s)^2.$$

This shows that

$$Q_N\approx e^{Nq(\beta J,\beta b,\mu_s)}\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^\infty d\mu e^{-\frac{NJ\beta}{2}[1-J\beta(1-\mu_s^2)](\mu-\mu_s)^2}$$

i.e.

$$Q_N\approx e^{Nq(\beta J,\beta b,\mu_s)}\sqrt{\frac{1}{1-J\beta(1-\mu_s^2)}}$$

Note that the $N$ factor is completely removed after integration and remains just into the argument of the exponential. Eq. (16-18) follow straightforwardly.

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