\begin{equation*}
\int \frac{dx}{2x+1}
\end{equation*}
\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*}