Answer
$\dfrac{1}{2}\ln (\dfrac{17}{8})$
Work Step by Step
Given: $\int^{\ln 2}_{0} \tanh 2x dx$
This can be re-written as:$\int^{\ln 2}_{0} \dfrac{\sinh 2x}{\cosh 2x} dx$
Plug in: $\cosh 2x=t$ and $dt=2 \sinh 2x dx$
Thus, the limits of integration will also get changed.
Then, $\int^{\ln 2}_{0} \dfrac{\sinh 2x}{\cosh 2x} dx=\dfrac{1}{2}\int^{17/8}_{1} \dfrac{dt}{t} =\dfrac{1}{2}[\ln |t|]^{17/8}_{1}=\dfrac{1}{2}[\ln \dfrac{17}{8}-\ln(1)] $
or, $=\dfrac{1}{2}\ln (\dfrac{17}{8})$
