The general expression of FM signal is

\(S\left(t\right)=E_csin\left(\omega_ct+\theta\right)\)

\(S\left(t\right)=E_csin\left(\omega_ct+2\pi k_f\int_{0}^{t}{e_mdt}\right)\)

Where

\(\theta=2\pi k_f\int_{0}^{t}{E_mcos\omega_mtdt}\)

\(S\left(t\right)=E_csin\left(\omega_ct+2\pi k_f\int_{0}^{t}{E_mcos\omega_mtdt}\right)\)

\(S\left(t\right)=E_csin\left(\omega_ct+\frac{2\pi E_mk_f}{\omega_m}sin\omega_mt\right)\)

\(S\left(t\right)=E_csin\left(\omega_ct+\frac{2\pi E_mk_f}{2\pi f_m}sin\omega_mt\right)\)

\(S\left(t\right)=E_c\sin\left(\omega_ct+\frac{∆f}{f_m}\sin\omega_mt\right)\)

Where \(E_mk_f=∆f\)

\(k_f\) is the frequency sensitivity.

∆f is the frequency deviation

\(\frac{∆f}{f_m}=\beta\)

\(\beta\) is called modulation index

The resultant FM signal expression is

\(S\left(t\right)=E_csin\left(\omega_ct+\beta sin\omega_mt\right)\)