## Thomas' Calculus 13th Edition

$\sqrt 3 \ln | \sinh (\dfrac{\theta}{\sqrt 3})| +C$
As we are given that $\int \coth (\dfrac{\theta}{\sqrt 3}) d\theta$ Now, plug $\dfrac{\theta}{\sqrt 3}=t$ and $d \theta =\sqrt 3 dt$ This implies: $\int \coth (\dfrac{\theta}{\sqrt 3}) d\theta=\sqrt 3 \int \coth t dt=\sqrt 3 \int \dfrac{\cosh t}{\sinh t} dt$ Now, consider $\sinh t =u$ and $\cosh t dt =du$ Thus, $\sqrt 3 \int \dfrac{\cosh t}{\sinh t} dt=\sqrt 3 \int (\dfrac{du}{u})= \sqrt 3 [\ln |u|] +C= \sqrt 3 \ln | \sinh (\dfrac{\theta}{\sqrt 3})| +C$