Answer
$$
\int \sqrt{e^{2 x}-1} d x=\sqrt{e^{2 x}-1}-\cos ^{-1}\left(e^{-x}\right)+C
$$
Work Step by Step
$$
\int \sqrt{e^{2 x}-1} d x
$$
If we make the substitution
$$
u=e^{x}, \quad \text { . Then } x=\ln u, \quad d x=d u / u,
$$
and we look at the Table of Integrals , we see that the closest entry is number $41$with $a=1$:
\begin{aligned}
\int \sqrt{e^{2 x}-1} d x &=\int \frac{\sqrt{u^{2}-1}}{u} d u \\
&\stackrel{41}{=} \sqrt{u^{2}-1}-\cos ^{-1}(1 / u)+C\\
&=\sqrt{e^{2 x}-1}-\cos ^{-1}\left(e^{-x}\right)+C
\end{aligned}
