Question

$\int \frac{e^{x}dx}{\sqrt{e^{2x}-1}}$

Answer 1

We have,

$\int \frac{e^x}{\sqrt{e^{2x}-1}}dx$

Let $e^x=t$

On differentiation and we get,

$e^{x}dx=dt$

Then,

$\int \frac{dt}{\sqrt{t^2-1}}$

$\Rightarrow \log |t+\sqrt{t^2-1}|+C$

Put $t=e^x$

$=\log |e^x+\sqrt{e^x-1}|+C$

Hence, this is the answer.