Answer
$\ln (\dfrac{5}{2})$
Work Step by Step
As we are given that $\int^{\ln 4}_{\ln 2} coth x dx$
Re-write:$\int^{\ln 4}_{\ln 2} \dfrac{\cosh x}{\sinh x} dx$
Now, consider $\sinh x=t \implies dt=\cosh x dx$
Thus, $\int^{\ln 4}_{\ln 2} \dfrac{\cosh x}{\sinh x} dx=\int^{15/8}_{3/ 4} (\dfrac{dt}{t}) =[\ln |t|]^{15/8}_{3/ 4}=\ln (\dfrac{15}{8})-\ln (\dfrac{3}{4})=\ln (\dfrac{15}{8} \cdot \dfrac{4}{3})$
Hence, $\int^{\ln 4}_{\ln 2} coth x dx=\ln (\dfrac{5}{2})$
