Integration of x^n*e^x/(e^x + 1)^2

[Littlepig]

I there.

I'm currently using this kind of integrals, with n even, and I couldn't found anything in internet for calculate this.
Let From the book I'm studying Ashcroft/Mermin, Solid State Physics, Append C, it says that
$$a_{n}=\int_{-\infty }^{\infty } \frac{x^n<br /> e^x}{\left(e^x+1\right)^2} \, dx$$

can, by elementary operations, be written as

$$a_{n}=1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}-\frac{1}{4^{2n}}+...$$ and so can be written with the zeta function:

$$a_{n}=(2-\frac{1}{2^{2(n+1)}}) \zeta(2n)$$

and, $$\zeta(2n)=2^{2n-1}\frac{\pi^{2n}}{(2n)!}B_{n}$$ where B_n are the bernoulli numbers.

Well, aren't any easier way, using integration in complex plane? Can you give me an ideia of where can I find a resolution to this?

Thanks,
littlepig

 

[Count Iblis]

You can do it in a similar way as in this case:

http://en.wikipedia.org/wiki/Stephan-boltzmann_law#Appendix