Answer
$\ln (\dfrac{5}{2})$
Work Step by Step
Given: $\int^{\ln 4}_{\ln 2} coth x dx$
This can be re-written as:$\int^{\ln 4}_{\ln 2} \dfrac{\cosh x}{\sinh x} dx$
Plug in: $\sinh x=t$ and $dt=\cosh x dx$
Thus, the limits of integration will also get changed
Then, $\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} $
or, $=\ln (\dfrac{15}{8} \cdot \dfrac{4}{3})$
or, $=\ln (\dfrac{5}{2})$
