Integral of $\operatorname{arcsinh}(\frac{x}{a})$, constant of integration

Question

I have been able to show using trigonometric substitution that

$$\int \operatorname{arcsinh}\left(\frac{x}{a}\right)\,dx = \ln\left(x+\sqrt{x^2+a^2}\right)-\ln(a)+C$$

However, when I go online to check the answer I notice that the answer is $$\ln\left(x+\sqrt{x^2+a^2}\right)+C$$ without $\ln(a)$. How do they get this answer? Do they consolidate $-\ln(a)+C$ into a new constant of integration?

Answer 1

You are finding an indefinite integral and $\ln(a)$ is a constant. Thus with the constant $C$ there is not need for another constant.

$C- \ln(a)$ is a constant and we might simply drop the $\ln(a)$ part.