# Thread: Integral of x/[1+exp(x)] dx

1. ## Integral of x/[1+exp(x)] dx

$\int_0^{10}(\frac{300x}{1+e^x})dx$

2. Originally Posted by ragungs
$\int_0^{10}(\frac{300x}{1+e^x})dx$
There is no closed form expression for this integral. Are you looking for an approximation?

-Dan

3. I need the true value of this integral to calculate the error of my approximation.

I wonder if this integral can be solved by hand.

4. Originally Posted by ragungs
I need the true value of this integral to calculate the error of my approximation.

I wonder if this integral can be solved by hand.
As I said there is no closed form expression. You cannot get an exact value of it.
$\int \frac{x}{1 + e^x}~dx = \frac{x^2}{2} - ln(1 + e^x) - \sum_{n = 1}^{\infty} \frac{e^{nx}}{n^2}$

There is no expression for that summation in terms of elementary functions. This thing can only be approximated.

-Dan

Edit: A made a mistake in copying "polylog" function from the Integrator site. I have fixed it.

5. $\int_0^{10} \left(\frac{300x}{1 + e^x}\right)dx \approx 246.59029$

I used my graphing calculator to calculate the integral... It's not exact either, but it's closer than a lot of people can get.

6. ok thanks all.

7. Originally Posted by topsquark
There is no closed form expression for this integral. Are you looking for an approximation?

-Dan
It does in terms of the polylogarithm ${\rm Li}_2$

RonL

8. Originally Posted by CaptainBlack
It does in terms of the polylogarithm ${\rm Li}_2$

RonL
$-Li_2(1+e^{x})-x\ln(1+e^{x})+\frac{1}{2}x^2$?