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

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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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