A biologist grows a culture of bacteria in a flask…

A biologist grows a culture of bacteria in a flask. She counts the population $N$ in thousands, and $t$ is the time in hours. From her measurements the rate of growth is $\dfrac{dN}{dt} = \dfrac{2t + 3}{t^2 + 4t + 7}$ thousand per hour. At $t = 0$ there are $2$ thousand bacteria, so $N(0) = 2$. She wants a formula for $N(t)$. The top is not the derivative of the bottom, which is $2t + 4$, so she splits the top as $2t + 3 = (2t + 4) - 1$. The first part gives a logarithm. For the second part she completes the square, $t^2 + 4t + 7 = (t + 2)^2 + 3$, and uses the inverse tangent. She writes $N(t) = \ln(t^2 + 4t + 7) + c\tan^{-1}\left(\dfrac{t + 2}{\sqrt{3}}\right) + K$. Find the number $c$. Then find $K$ from $N(0) = 2$, and find the population $N(2)$ after two hours. Give $c$ to four decimal places, and $K$ and $N(2)$ in thousands to four decimal places.