representing arcsinh as a logarithm I am trying to understand this equality:
$$ \ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C= \ln{|x+\sqrt{x^2+4}|} + C'$$
My teacher didn't really explain it, she just noted that "the difference between the two statements is a constant (This equality is an answer for an integral so she just changed $C$ to $C'$).
Can anyone please explain it? Thanks!

$$\ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C=\ln{\left|\frac{x}{2}+\sqrt{\frac{x^2+4}{4}}\right|}+C$$
$$=\ln{\left|\frac{x+\sqrt{x^2+4}}{2}\right|}+C$$
$$= \ln{|x+\sqrt{x^2+4}|} -\ln2+ C$$
$$= \ln{|x+\sqrt{x^2+4}|}+ C'.$$
