Relationship between modulation index and carrier-suppressed condition in fm

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?

It seems an absolute pull range of a VCXO will limit the modulation index. Is there possibly a way to extend it?

You could use a colpitts oscillator and varactor - you could "wrap" around it a PLL so that centre-frequency stability is as good as a reference crystal. I've done this a few times on applications ranging from 80 MHz to 400 MHz and got FM deviations of +/- 5MHz no problem and data rates of up to 15 MBpsec.

EDIT

Here's a PLL that takes 6mA and works upto centre frequencies of 400MHz: -

You need a small MCU for setting up the division ratio to make lock plus a simple varactor controlled colpitts will do the trick. Or, try the LTC 6905: -

I particularly like this device because it's really simple and doesn't go belly-up on large G-forces (unlike cheap xtals)