What is the integral of dx/((e^x)+1)^(1/2)?

Question

The answer that I need to get to is ln[((1+(e^x))^(1/2) - 1) / ((1+(e^x))^(1/2) +1)].  Any help here would be great cause I have no idea

Hemant · Accepted Answer

I = ∫ [ 1 / √(1 + ℯˣ) ] dx ................................ (1)
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Let : u = √(1+ℯˣ).

Then : u² = 1 + ℯˣ

2u· du/dx = ℯˣ = u² - 1, i.e., dx = 2u· du / (u²-1).
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From (1), then,

I = ∫ ( 1/u ) • 2u· du / (u²-1)

= ∫ [ 2 / (u²-1) ] du

= ∫ [ 2 / ((u-1)(u+1)) ] du

= ∫ { [ 1 / (u-1) ] - [ 1 / (u+1) ] } du

= ln | u-1 | - ln | u+1 | + C

= ln | (u-1) / (u+1) | + C

= ln | [ √(1 + ℯˣ) - 1 ] / [ √(1 + ℯˣ) + 1 ] | + C ........... Ans. 
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ros · Answer

Not according to:

http://www.wolframalpha.com/input/?i=What+is+the+integral+of+dx%2F%28%28e%5Ex%29%2B1%29%5E%281%2F2%29%3F

But maybe I don't understand your question.

panic mode · Answer

http://www.wolframalpha.com/input/?i=integral+of+dx%2F%28%28e%5Ex%29%2B1%29%5E%281%2F2%29

click on "show steps" if you want details

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What is the integral of dx/((e^x)+1)^(1/2)?

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