How to Solve the Integral: Evaluate ∫(1+e^-x)^1/2 / e^x from 0 to 1

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SUMMARY

The integral ∫(1+e^-x)^(1/2) / e^x from 0 to 1 can be effectively solved using u-substitution and integration by parts. The key substitution is u = e^-x + 1, which simplifies the integrand significantly. By rewriting e^x in the denominator as e^-x times the square root expression, the integral becomes manageable. The discussion concludes with the successful evaluation of the integral, demonstrating the effectiveness of these techniques.

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Homework Statement



Hi guys, so I've been having some trouble with this specific integral, and would like some help on how to solve it.\int^1_0\sqrt{1+e^{-x}}/e^x.

2. The attempt at a solution

Editing

Next stage I'm sort of confused because I've never encountered two variables with "u".
 
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With you u substitution, all you have done is replace 'x' with 'u'. You haven't done any essential simplification to the integrand.

I would suggest looking at integration by parts. If you can find candidates for u and v with parts, then further substitution might be warranted.
 
SteamKing said:
With you u substitution, all you have done is replace 'x' with 'u'. You haven't done any essential simplification to the integrand.

I would suggest looking at integration by parts. If you can find candidates for u and v with parts, then further substitution might be warranted.

Yeah I wasn't sure if I should substitute more lol or just go through with algebra. Thanks.
 
Actually usub does work. Re write e^x in the denominator as e^-x times the squareroot expression. u=e^-x+1. does it looks like du could be written in terms of e^-x?
 
I figured it out

IT HAS BEEN SOLVED :)!
 
I am Batman.
 

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