Answer
\[\frac{x}{2}+\frac{7}{4}\ln |2x-1|+C\]
Where $C$ is constant of integration
Work Step by Step
Let \[I=\int\frac{x+3}{2x-1}dx\]
\[I=\frac{1}{2}\int\frac{(2x-1)+7}{2x-1}dx\]
\[I=\frac{1}{2}\int dx+\frac{7}{2}\int\frac{dx}{2x-1}\]
Let \[I_1=\int\frac{dx}{2x-1}\]
Put $t=2x-1$
\[\Rightarrow dt=2dx\]
\[I_1=\frac{1}{2}\int\frac{dt}{t}\]
\[I_1=\frac{1}{2}\ln |t|\]
\[I_1=\frac{1}{2}\ln |2x-1|\;\;\;...(2)\]
Using (2) in (1)
\[I=\frac{x}{2}+\frac{7}{4}\ln |2x-1|+C\]
Where $C$ is constant of integration
Hence ,
\[I=\frac{x}{2}+\frac{7}{4}\ln |2x-1|+C.\]
