$ \int \frac{dx}{x+3}$ = "$\ln{|x+3|} + C$" or "$\ln{(x+3)} + C$"?

Question

$ \int \frac{dx}{x+3}$ = "$\ln{|x+3|} + C$" or "$\ln{(x+3)} + C$" ?

I don't know if the module is needed in the answer record.

Answer 1

A function $f:\Bbb R\setminus \{-3\}\to \Bbb R$ is an antiderivative of $\frac1{x+3}$ if and only if there are two constants $C_1$, $C_2$ such that $$f(x)=\begin{cases}\ln(x+3)+C_1&\text{if }x>-3\\ \ln(-x-3)+C_2&\text{if }x<-3\end{cases}$$