# How do you integrate e^x / sqrt(1-e^(2x)) dx?

$\arcsin \left({e}^{x}\right) + C .$
Let ${e}^{x} = t$, so that, ${e}^{x} \mathrm{dx} = \mathrm{dt} .$ Also, note that, ${e}^{2 x} = {t}^{2.}$
Hence, $I = \int {e}^{x} / \sqrt{1 - {e}^{2 x}} \mathrm{dx} = \int \frac{1}{\sqrt{1 - {t}^{2}}} \mathrm{dt} = \arcsin t = \arcsin \left({e}^{x}\right) + C .$