Integrating the Inverse Exponential-Square Root Function

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SUMMARY

The forum discussion centers on solving the integral \(\int \frac{dx}{e^{x}\sqrt{1-e^{-2x}}}\). Participants suggest using the substitution \(u = e^{-x}\) to simplify the integral, which transforms it into a more manageable form. The discussion concludes with two potential solutions: \(\cos^{-1}(e^{-x}) + C\) and \(-\sin^{-1}(e^{-x}) + C\), indicating that both forms are valid answers to the integral.

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Homework Statement


\int dx/(e^{x}\sqrt{1-e^{-2x}})

Homework Equations





The Attempt at a Solution


I have absolutely no idea of how to start the problem, any help is greatly appreciated!
thanks!
 
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Try a substitution of maybe u=e-x. Then you should get it into a form where the anti-derivative should be easily found.
 
Also, note that 1/ex = e-x.
 
hmm, can i get a little more hint? if i do a u sub. of u = e^{-x}, how do i get rid of
e^{-2x}? I have tried do a u sub. of u = \sqrt{1-e^{-2x}}, but i ended up getting the \int du/e^{-x}, which i don't know how to proceed after this. What am i doing wrong?
 
e-2x=(e-x)2, so in terms of u it is?
 
o boy, that was embarrassing lol
just want to make sure, is the answer cos^{-1}(e^{-x}) + C?
 
That should be correct.

I think -sin-1(e-x)+C should work as well.
 

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