How Do You Integrate e^-x / (1 + e^-x) dx?

In summary, the integral of (e^-x)/(1+e^-x)dx can be solved using u-substitution by letting u = 1+e^-x and finding the value of du.
  • #1
menco
43
0

Homework Statement


(e^-x)/(1+e^-x)dx


Homework Equations


Integral of e^x function


The Attempt at a Solution


I am completely lost with the problem, the fact that it is e^-x and not e^x has me stupped as well as the division.
 
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  • #2
Try [itex]u[/itex]-substitution.
 
  • #3
so if I let u=1+e^-x?
 
  • #4
yes?
 
  • #5
menco said:
so if I let u=1+e^-x?

Well, does it work?
 
  • #6
I still can't figure it out so I gave up and will try again later
 
  • #7
menco said:
I still can't figure it out so I gave up and will try again later

When you do get back to it, answer this: What is [itex]du[/itex] when [itex]u=1+e^{-x}[/itex]?
 
  • #8
Will du = -e^-x dx
 

1. What does the integral of e^-x/(1+e^-x)dx represent?

The integral of e^-x/(1+e^-x)dx represents the area under the curve of the function e^-x/(1+e^-x) from 0 to infinity.

2. How do you solve the integral of e^-x/(1+e^-x)dx?

The integral of e^-x/(1+e^-x)dx can be solved using the substitution method, where u = 1+e^-x and du = -e^-x dx. This will simplify the integral to -1/u du, which can then be solved using the natural logarithm function.

3. Can the integral of e^-x/(1+e^-x)dx be solved using any other methods?

Yes, the integral can also be solved using integration by parts or the partial fractions method.

4. Is there a specific domain for which the integral of e^-x/(1+e^-x)dx is valid?

The integral of e^-x/(1+e^-x)dx is valid for all real numbers.

5. Is there a physical interpretation of the integral of e^-x/(1+e^-x)dx?

Yes, the integral can represent the average value of a function over a given interval or the expected value of a random variable in a certain distribution.

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