Answer
\[\frac{1}{2}\ln \left(\frac{11}{5}\right)\]
Work Step by Step
Let \[I=\int_{2}^{5}\frac{dr}{1+2r}\]
\[\Rightarrow I=\left[\frac{1}{2}\ln |1+2r|\right]_{2}^{5}\]
\[\Rightarrow I=\left[\frac{1}{2}\ln |1+2(5)|-\frac{1}{2}\ln |1+2(2)|\right]\]
\[\Rightarrow I=\frac{1}{2}\ln 11-\frac{1}{2}\ln 5\]
\[\Rightarrow I=\frac{1}{2}\ln \left(\frac{11}{5}\right)\]
Hence , \[\int_{2}^{5}\frac{dr}{1+2r}=\frac{1}{2}\ln \left(\frac{11}{5}\right)\]
