Specifically, when integrating $\frac{1}{ax+b}$ we get $\frac{1}{a}\ln|ax+b|$.
However, if we multiply the integrand by say $c/c = 1$, then the integral computes to $(1/a)\ln|c(ax+b)|$. Can someone please explain why this is so or where my error may be?
Thanks.
We have $$\int\frac{1}{ax+b} \, \mathrm{d}x = \frac{1}{a}\ln{|ax+b|} + k$$
Whilst $$\int \frac{1}{ax+ b} \cdot \frac{c}{c} \, \mathrm{d}x = \frac{1}{a}\ln{ |c(ax+b)|} + k' = \frac{1}{a}\ln |c| + \frac{1}{a}\ln{|ax+b|} + k'$$
We need only take $k = k' + \frac{1}{a}\ln |c|$ to verify that both solutions are identical up to an additive constant.