Electronic – Is the value Vm correct in a FM modulation

The instantaneous frequency of a signal \$A\cos(\phi(t))\$ where \$\phi(t)\$ is an arbitrary function of time is defined as the derivative of \$\phi(t)\$ if you want to measure frequency in radians per second and as \$\frac{1}{2\pi}\$ times the derivaive_ of \$\phi(t)\$ if you want to measure frequency in Hertz. Of course, in the common case of a fixed frequency this corresponds to the familiar \$\phi(t) = \omega_c t+\phi_0 = 2\pi f_c t + \phi_0\$.

The standard definition of a frequency-modulated signal is one in which the deviation of the instantaneous frequency (at time \$t_0\$, say), from the carrier frequency is proportional to the value \$x_m(t_0)\$ of the modulating signal \$x(t)\$ at time \$t_0\$. The constant of proportionality is denoted by \$f_{\Delta}\$ in your notation: a \$1\$ volt signal creates a deviation of \$f_{\Delta}\$ Hz. Thus, if \$A\cos(\phi(t))\$ is the FM signal, then we have that $$ \left.\frac{\mathrm d}{\mathrm dt}\phi(t)\right|_{t=t_0} = 2\pi f_c + 2\pi f_{\Delta} x_m(t_0) $$ so that the deviation of the instantaneous frequency \$f_c + f_{\Delta} x_m(t_0)\$ from the carrier frequency \$f_c\$ is \$f_{\Delta} x_m(t_0)\$, just as we want it to be. It then follows from the fundamental theorem of calculus that $$\phi(t_0) = \int_{0}^{t_0}2\pi f_c + 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt = 2\pi f_c t_0 + \int_{0}^{t_0} 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt$$ or, with a slight change in notation, the FM signal can be expressed as $$A\cos\left(2\pi f_c t + \int_{0}^{t} 2\pi f_{\Delta} x_m(\tau)\, \mathrm d\tau\right)$$ the way you have it. Note that \$A\$ is the amplitude of the FM signal and is fixed; it is the frequency that is varying. Surely we need to distinguish between the FM signal when it is created using a voltage-controlled oscillator with an amplitude of \$1\$ volt and when it comes out of the power amplifier and goes to the antenna with a power of 10 kW?

To reconcile your (correct) understanding about modulation with the book, let me restate

modulation modifies a signal in a way that is able to carry information

as

modulation modifies a carrier signal in a way that is able to carry an information signal

Now in PCM, the bitstream is the carrier signal, and the "analog signal" he refers to is the information signal. Does this make it clearer?