Same integration question yields to 2 possible answers; why? In looking at the equality
$$\int \frac{a}{b(c-x)}dx = \int dt$$
I obtained different answers via different methods.
Via one method, I got
$$- \frac{a}{b} \ln(c-x) = t+C$$
Via another, I got
$$- \frac a b \ln(b(c-x)) = t+C$$
How is this possible?

The $C$ in both answers are not the same, assuming $a,b,c$ are constants.
Note that
$$-\frac a b \ln(b(c-x)) = - \frac{a}{b} \ln(b) - \frac a b \ln(c-x) = t + K$$
If we set $K = C - a \ln(b) / b$, then it becomes equivalent to your first result.
