I got this from my workbook solution, was able to solve the question for the most part but stuck in the last sentence.
$$7(\ln\left|x+\sqrt{x^2-7}\right|-\ln\sqrt{7}) + c = 7 \ln\left|x+\sqrt{x^2-7}\right| + c$$
Does this means that $-\ln\sqrt{7} = 0$? but when I use a calculator it doesn't give me $0$.
The original question is $\int \frac{7}{\sqrt{x^2-7}} \,dx$.
It just got absorbed into the $C$. Since $C$ is any constant, $C'=C-7\ln(\sqrt{7})$ works just as well. The author is just being sloppy in calling them the same thing.