What exactly is "integrated form"? I am reading on population growth and I see

$\Delta N_t = (b - d)N_t \, \Delta t = mN_t \, \Delta t$ ,
where $m = b - d$.
As $\Delta t \to 0$, this becomes
$\dfrac{dN_t}{dt} = mN_t$.
In integrated form, $mt = \log_eN_t - \log_eN_0$, or
$N_t = N_0e^{mt}$

I think I understand that $dN_t/dt$ is the rate of population growth at time $t$, and that $N_t = N_0e^{mt}$ gives the population at time $t$ as an exponential function of $N_0$ starting population, $m$ growth rate, and $t$ time. But what does "integrated form" mean, and what are the steps that get us from $dN_t/dt = mN_t$ to $N_t = N_0e^{mt}$?
 A: The equation
$$
\frac{dN_t}{dt} = m N_t \tag 1
$$
is a differential equation.  It says something about $N_t$ but it does not say
$$
N_t = \text{some explictly specified function of }t. \tag 2
$$
Going from $(1)$ to $(2)$ is called solving the differential equation.  This is done by integrating:
\begin{align}
\frac{dN_t}{dt} & = mN_t \\[8pt]
\frac{dN_t} {N_t} & = m\,dt \\[10pt]
\int \frac{dN_t} {N_t} & = \int m\,dt \\[10pt]
\log |N_t| & = mt+\text{some constant} \\[10pt]
|N_t| & = e^{mt+\text{constant}} = e^{mt}\cdot\text{some positive constant} \\[10pt]
N_t &  = e^{mt}\cdot\text{some constant}
\end{align}
When $t=0$ then $N_t = N_0$ but $N_t=e^{m\cdot0}\cdot\text{constant}= 1\cdot\text{constant}$, so the "constant" is $N_0$. 
Now let's go back to the integration step: It is a bit of an abuse of notation to write $\displaystyle\int_{N_0}^{N_t}\frac{dN_t}{N_t}$ since we're using the notation $N_t$ to refer to two different things.  This evaluates to
$$
\log N_t - \log N_0.
$$
The same abuse of notation happens if we write $\displaystyle\int_0^t m\,dt$, and it comes to $mt$.  Thus integrating is what gives us the equality
$$
\log N_t-\log N_0=mt.
$$
Since it came from integrating, it is getting called an "integrated" form.
