The difficulty with respect to this question as well as a related one is that the authors of the text that the OP is using have chosen to adopt a definition of an FM signal that is not
the one used by most other writers. This unfortunate choice muddies
the study of what is already a difficult-to-understand modulation method
(as compared to amplitude modulation and even phase modulation). Most people
use take an FM signal to be of the form given in
Wikipedia where,
for a modulating signal $x(t)$ (assumed to be of magnitude at most $1$), the FM signal is
of the form
$$A \cos\left(2\pi f_c t + 2\pi f_\Delta \int_0^t x(\tau)\,\mathrm d\tau \right)$$
where $f_c$ is the carrier frequency and $f_\Delta$ is the maximum frequency deviation from the carrier.
The instantaneous frequency is the derivative of the phase (argument of the $\cos$) with respect to time $t$, and is thus
$$\begin{align*}
\omega_{\scriptstyle{\text{inst}}}
&= \frac{\mathrm d}{\mathrm dt}\left[2\pi f_c t
+ 2\pi f_\Delta \int_0^t x(\tau)\,\mathrm d\tau \right]
= 2\pi f_c + 2\pi f_\Delta x(t) ~ \text{rad/sec},\\
f_{\scriptstyle{\text{inst}}}
&= \frac{1}{2\pi}\frac{\mathrm d}{\mathrm dt}\left[2\pi f_c t
+ 2\pi f_\Delta \int_0^t x(\tau)\,\mathrm d\tau \right]
= f_c + f_\Delta x(t) ~ \text{Hz}.
\end{align*}$$
When $x(t)$ is a sinusoidal signal, say $\cos(2\pi f_s t)$,
its integral is
$$\int_0^t \cos(2\pi f_s \tau)\,\mathrm d\tau
= \left.\frac{1}{2\pi f_s}\sin(2\pi f_s\tau)\right\vert_0^t
= \frac{1}{2\pi f_s}\sin(2\pi f_s t)$$
is also a sinusoid at the same frequency, and so for
this special case, the FM signal can be written as
$$A \cos\left( 2\pi f_c t + \frac{f_\Delta}{f_s} \sin(2\pi f_s t)\right)
= A \cos\left(2\pi f_c t + k\sin(2\pi f_s t)\right).$$
Note that the instantaneous frequency of the FM signal
varies (in sinusoidal fashion) between $f_c-k$ and $f_c+k$ Hz.
Note also the difference between this standard result
and what the OP claims is the formula in his textbook
(I have not read the text myself and have no idea what
it actually says):
$$v(t) = 8 cos (2\pi [10^5 + 4 sin(5 \times 10^3 t)]\,\mathbf t\,)$$
The extra $\mathbf t$ at the end changes the instantaneous frequency
to
$$2\pi10^5+8\pi sin(5 \times 10^3t) + 8 \pi \, \mathbf{t}\, (5 \times 10^3) cos (5 \times 10^3 t)$$
(as correctly calculated by the OP). But,
instead of the instantaneous frequency fluctuating between
fixed limits, the maximum frequency deviation increases with time.
The claim that this is an FM signal (in the
usual sense of the word) is certainly not tenable.
If OP Samuel has correctly copied the problem from
his textbook, then the problem, as stated, has no solution.
Perhaps it is just a typo in this particular problem.
On the other hand,
if the same "additional" $t$ occurs in
occurs in the text in the definition of an FM signal,
and this formulation is used throughout the book, then
the book certainly has a major blunder in it.
