1. ## integral tables

I need to integrate this integral from -infinity to infinity,

∫▒〖(x^3 e^x)/(e^x+1)^2 dx〗

The question says I can look it up in integral table but I cannot find tables with this type of integral in it.

2. note that it is an odd function ( a fantastic odd function )

3. So it cancels out to 0?

If that is the case then I need to integrate (x^4*e^x)/((e^x+1)^2).

I am doing a sommerfield expansion and I need another term to survive.

4. $\displaystyle I = \int_{-\infty}^{\infty} \frac{ x^{2n} e^x }{ ( 1 + e^x )^2 } ~dx = 2(2n)! \zeta(2n)( 1 - 2^{1-2n} )$

when $\displaystyle n = 2$

$\displaystyle I = 2(4)! \zeta(4) ( 1 - 2^{-3} ) = \frac{7}{15} \pi^4$

5. Perfect. Thank You.

6. Originally Posted by simplependulum
note that it is an odd function ( a fantastic odd function )
I haven't checked the details here but care must be taken with improper integrals of this type. An odd integrand does not mean that an improper integral of this type is equal to zero. It could be divergent. A case in point is $\displaystyle \int_{-\infty}^{+\infty} \frac{x}{1 + x^2} \neq 0$ ....

7. Originally Posted by mr fantastic
I haven't checked the details here but care must be taken with improper integrals of this type. An odd integrand does not mean that an improper integral of this type is equal to zero. It could be divergent. A case in point is $\displaystyle \int_{-\infty}^{+\infty} \frac{x}{1 + x^2} \neq 0$ ....

Oh , I didn't realise this ... but luckily this integral $\displaystyle \int_0^{\infty} ~(...)~dx$ mentioned above is convergent.