One way to compute $\int \frac{dx}{a-x}$ for $a>0$ is $- \int \frac{d(a-x)}{a-x} = - \ln |a-x|$; another way is factoring out the $a$ in the denominator first: $$ \int \frac{\frac{1}{a}}{1 - \frac{x}{a}} dx = - \int \frac{d \Big(1 - \frac{x}{a}\big)}{1 - \frac{x}{a}} = - \ln |1 - \frac{x}{a}|. $$ While it is understandable that two anti-derivatives can differ by a constant, which, in this case, is $\ln(a)$, how to understand that an equivalent algebraic operation leads to two different results? Where did I go wrong?
Thanks!
