From the Wikipedia page on chirp, if I provide some time-dependent frequency function $f(t)$, this has to be integrated if I want to use it as the argument to, say, a sinusoid. Suppose $f(t) = f_0+kt$; then
$x(t) = \sin\left(2\pi\int_0^t f(\tau)d\tau\right) = \sin\left[2\pi\left(f_0t+\frac{k}{2}t^2+\phi_0\right)\right]$
Why is it not good enough to claim that $x(t) = \sin(2\pi f(t)t) \equiv \sin(\omega(t)t)$?