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

Question

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

I tried to solve by using integration by parts, but I couldn't find a solution. What method should I use to integrate this?

Answer 1

Hint: Let $u=1-e^{x}$${}{}{}{}{}{}{}$

Answer 2

$$ \int\frac{e^x}{\sqrt{1-e^x}}\underbrace{\Big(e^x \, dx\Big)}_{\text{HINT}} $$

After that substitution (the one that is hinted at), use another substitution, namely a rationalizing substitution.