Two different solutions to integral Given the very simple integral
\begin{equation}
\int -\frac{1}{2x} dx
\end{equation}
The obvious solution is
\begin{equation}
\int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C
\end{equation}
However, by the following integration rule
\begin{equation}
\int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C
\end{equation}
the following solution is obtained
\begin{equation}
\int -\frac{1}{2x} dx = -\frac{1}{2}\ln{|-2x|} + C
\end{equation}
Why are these solutions different? Which is correct? 
The second solution can be simplified
\begin{equation}
-\frac{1}{2}\ln{|-2x|} + C = -\frac{1}{2}\ln{|-2|} -\frac{1}{2}\ln{|x|} + C= -\ln{\frac{1}{\sqrt{2}}} - \frac{1}{2}\ln{|x|} + C
\end{equation}
but they still differ.

Both of the results is Ok. Note that there is no need that two constants $C$ in the first result and $C$ in the second one are the same. Here, we have $C_1=C_2\times0.5\ln|2|$.
