integral tables

**thealbino** · December 12th 2009, 05:54 PM

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.

**simplependulum** · December 12th 2009, 07:38 PM

note that it is an odd function ( a fantastic odd function

)

**thealbino** · December 12th 2009, 08:28 PM

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.

**simplependulum** · December 12th 2009, 08:58 PM

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

when $n = 2$

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

**thealbino** · December 12th 2009, 09:08 PM

Perfect. Thank You.

**mr fantastic** · December 13th 2009, 04:22 AM

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 $\int_{-\infty}^{+\infty} \frac{x}{1 + x^2} \neq 0$ ....

**simplependulum** · December 14th 2009, 01:47 AM

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 $\int_{-\infty}^{+\infty} \frac{x}{1 + x^2} \neq 0$ ....

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

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