\begin{equation*}
\begin{split}
& u = 2x+1
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \frac{du}{dx} = 2 \\\\
& du = 2 \: dx \\\\
& dx = \frac{du}{2}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \int \frac{dx}{2x+1} \\\\
& \int \frac{1}{2x+1} \: dx \\\\
& \int \frac{1}{u} \: \frac {du}{2} \\\\
& \frac 12 \int \frac 1u \: du \\\\
& \frac 12 \ln u + C \\\\
& \bbox[5px, border: 2px solid magenta] {\frac{1}{2} \: \ln|2x+1| + C}
\end{split}
\end{equation*}