Question

Help with statistical mechanics integral

Answer 1

\[\int\limits_0^{\infty} \frac{x^2 e^x}{1+e^{x}}dx\]

Answer 2

does it converge??

Answer 3

as suspected the integral does not converge
http://www.wolframalpha.com/input/?i=Integrate [x^2*(e^x%2F(1+%2B+e^x))%2C+{x%2C+0%2C+Infinity}]

Answer 4

I'm sorry, I wrote it in wrong:
\[\int\limits_0^{\infty} \frac{x^2 e^x}{(1+e^x)^2}dx\]

Answer 5

seems to have interesting result
http://www.wolframalpha.com/input/?i=Integrate%5Bx%5E2*%28e%5Ex%2F%281+%2B+e%5Ex%29%5E2%29%2C+%7Bx%2C+0%2C+Infinity%7D%5D

Answer 6

\[ \large \begin{array}{c l}\int_0^\infty x^2\frac{e^x}{(1+e^x)^2}dx & =\int_0^\infty x^2\sum_{n=1}^\infty n(-1)^{n-1}e^{-nx}dx \\ & = \sum_{n=1}^\infty n(-1)^{n-1}\int_0^\infty x^2e^{-nx}dx \\ & = \sum_{n=1}^\infty n(-1)^{n-1}\int_0^\infty \left(\frac{x}{n}\right)^2e^{-x}\frac{dx}{n} \\ & = \left(\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}\right)\int_0^\infty x^2e^{-x}dx \\ & = \left(1-\frac{2}{2^2}\right)\zeta(2)\Gamma(3) \\ & = \frac{\pi^2}{6}. \end{array} \]

Thanks to people in MSE

Answer 7

I need to learn techniques like that haha