Integral of Natural Logarithm

Question

I been trying to solve the following problem:

When I solve it myself, depending on which technique I use, I end up finding either

or

Yet both results are completely different. I even tried typing it on a calculator, and this is what I get.

Can someone please explain how one can proceed to go from the second to last step to the last step, because both of these last steps give different results?

Thank you for your help.

Answer 1

They are actually the same. This is because

$$\begin{equation}\begin{aligned} \frac{3\ln(|5x + 5|)}{5} + C & = \frac{3\ln(|5(x + 1)|)}{5} + C \\ & = \frac{3(\ln(|x + 1|) + \ln(5))}{5} + C \\ & = \frac{3\ln(|x + 1|)}{5} + \frac{3\ln(5)}{5} + C \\ & = \frac{3\ln(|x + 1|)}{5} + C_2 \end{aligned}\end{equation}\tag{1}\label{eq1A}$$

where $C_2 = \frac{3\ln(5)}{5} + C$ is also an arbitrary constant since $C$ is an arbitrary constant. In other words, you just have another constant to include in the value of the arbitrary constant $C$ in the last step.