How would you integrate the function 1/(e^x +1)?

Question

the function is one divided by the quantity of e to the x plus 1

Answer 1

∫dx/(e^x + 1)

=>∫(e^x + 1 - e^x) dx/(e^x+1)

=>∫(e^x+1)/e^x+1) dx - ∫e^x dx /(e^x+1)

=>∫dx - ∫d(e^x+1)/e^x+1

=>x - ln(e^x+1) + c

Answer 2

Try an anti-derivative approach. YOu know the answer has to contain a term that goes like:

ln(e^x +1) so ask what term you would add/subtract to that such that when you take a derivative you get the integrand.

So,

f(x) +a*ln(e^x+1) = F

dF/dx = 1/(e^x+1) = f'+a*e^x/(e^x+1) = (f' e^x +f' + a*e^x)/(e^x+1)

Then you get by cancelling denominators:

1 = e^x(f'+a)+f' since there is no x depedent terms on the left hand side, f' = -a and f' =1

hence a =-1 and f' = 1 so that f = x. SO the intergal evaluates to:

x - ln(e^x+1)

Answer 3

First you multiply the numerator and denominator by e^x... you can do that because e^x is different of zero.

You have now to integrate e^x / (e^x(e^x+1))

You substitute the variable u = e^x + 1 and du = e^xdx

Now you have to integrate du/ (u-1)u try into partial fractions

1/(u-1)u = A/(u-1) + B/u

Can you go on?

Answer 4

Let u = e^x + 1, then
du/dx = e^x = u - 1 and
dx = 1/(u-1) du, so

integral 1/(e^x + 1) dx = integral 1/(u*(u-1)) du
Now
1/(u*(u-1)) = 1/(u-1) - 1/u
so the u-integral gives

ln(u-1) - ln(u) + c
= ln(e^x) - ln(e^x+1) + c
= x - ln(e^x+1) + c.

Answer 5

We have 1/(e^x + 1) = (e^(-x))/(1 + e^(-x)) = - (-e^(-x))/(1 + e^(-x)). Now, the numerator is the derivative of the denominator, so that

Int 1/(e^x +1) dx = - ln(1 + e^(-x)) + C

You can easily chack that d/dx ( - ln(1 + e^(-x)) + C) = - (-e^(-x)/(1 + e^(-x)) = e^(-x)/(1+ e^(-x)) = 1/(e^x +1)

Answer 6

multiply both the numerator and denominator by e^(-x).

If you do this you'll get the integral of

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

which looks complicated but it's easy using the U substitution U = 1 + e^(-x).

Good Luck!!!

Answer 7

∫1/(e^x +1) dx
= ∫1/[e^x(e^x +1)] de^x, mental substitution
= ∫[1/e^x - 1/(e^x +1)] de^x
= x - ln(e^x +1) + c
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Ideas to save time: Use mental substitution.