Natural logarithm question 1

Question

i tried to derive logistic population model, and need to integrate this $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t$. here is my solution

$\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=\int \frac{1}{k-N_t} dN_t=-\int \frac{1}{k-N_t}d{(k-N_t)}=-\ln\mid k-N_t\mid+C_1$. i think i have done something wrong here, because if i solve it this ways $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=-\int \frac{1}{1-\frac{N_t}{k}} d(1-\frac{N_t}{k})=-\ln \mid 1-\frac{Nt}{k} \mid +C_2$ which is obviously different from the previous solution, so where is the mistake(s) ?

Answer 1

Notice that $$-\log\left(1-\frac1kx\right)=-\log\left(\frac1k\left(k-x\right)\right)=-\log(k-x)-\log\left(\frac1k\right),$$ and that $k$ is constant with respect to $x$, which shows that your antiderivatives differ by a constant, namely $-\log\left(1/k\right)$.

For more information look up the fundamental theorem of calculus.

Answer 2

Since

$$\frac1k=-\left(\frac{N_t}{k}\right)'\implies -\int\frac{-\frac1k}{1-\frac{N_t}k}dN_t=-\log\left(1-\frac{N_t}k\right)+C=\log\frac k{k-N_t}+C$$