quantum mechanics - Calculating the the kernel using path integrals for quadratic lagrangians

I am reading Feynman and Hibbs on Path Integrals. In section 3.5, they show that the kernel for a lagrangian of the form $L=a(t)\dot{x}^2+b(t)\dot{x}x+c(t)x^2+d(t)\dot{x}+e(t)x+f(t)$ is $K(b,a)=e^{\frac{i}{\hbar}S_{cl}[b,a]}F(t_a,t_b)$. In general, how do I calculate the factor $F(t_a,t_b)$. In the problems after the section, I have calculated the classical action, for the particle in a magnetic field, and the forced harmonic oscillator. But I don't know how to calculate the prefactors. For e.g. this is the problem 3-11 from Feynman and Hibbs asks you to calculate the kernel of the harmonic oscillator driven bby an external force $f(t)$. The Lagrangian is $L=\frac{m}{2}\dot{x}^2-\frac{m\omega^2}{2}x^2+f(t)x$. The answer is $$K=\sqrt{\frac{m \omega}{2 \pi i \hbar \sin{\omega T}}}e^{\frac{i}{\hbar}S_{cl}}$$

where $T=t_f-t_i$ and $S_{cl}$ is the classical action. How can I see that the above is the factor multiplying the exponent directly or via a calculation.