<s>[INST] Getting 2 different solutions to the integral of $\frac{dx}{2x}$ I get two different answers that seem to conflict.
Is there an error in one method??
Method 1
\begin{align}
\int \frac{\mathrm{d}x}{2x}&=\frac 1 2 \int\frac{\mathrm{d}x}{x}\\
&=\frac 1 2\ln|x|+C
\end{align}
Method 2
\begin{align}
\int \frac{\mathrm{d}x}{2x}&=\frac 1 2 \int \frac{2\mathrm{d}x}{2x}\\
&=\frac 1 2\int \frac{\mathrm{d}u}{u},\;\;\text{ where }u=2x,\ \mathrm{d}u=2\mathrm{d}x\\
&=\frac 1 2 \big(\ln |u| + C\big)\\
&=\frac 1 2 \ln|2x|+C
\end{align}
 [/INST] Hint: $\frac{1}{2}\ln{|2x|} + C = \frac{1}{2}\ln{|x|} + \frac{1}{2}\ln{2} + C = \frac{1}{2}\ln{|x|} + \left(\frac{1}{2}\ln{2} + C\right)$.
</s>