How Do You Integrate the Electron Heat Capacity Integral by Hand?

[Tphysics]

1. The answer to this problem is easy when plugged into mathematica it's (pi^2)/3. I am trying to integrate it by hand however and can't figure out how to start it. I also can't find any other attempts of it online (our professor says we can just look it up if we can find it).

[(x^2*E^x)/(E^x + 1)^2, {x, -Infinity, Infinity}]

2. No equations

3. I've tried U-sub with setting U= (e^x+1) and then tried some integration by parts but I'm not getting there.

 

[fzero]

1. The answer to this problem is easy when plugged into mathematica it's (pi^2)/3. I am trying to integrate it by hand however and can't figure out how to start it. I also can't find any other attempts of it online (our professor says we can just look it up if we can find it).

[(x^2*E^x)/(E^x + 1)^2, {x, -Infinity, Infinity}]

2. No equations

3. I've tried U-sub with setting U= (e^x+1) and then tried some integration by parts but I'm not getting there.

This actually turns out to be very complicated to do and I am having trouble giving hints that you can follow without giving too much of the answer away, so please bear with me. At least using Mathematica seems like a legitimate solution to the problem and I don't believe that many people would expect an undergrad to come up with the solution below on their own.

First, integrals of functions of $x^n$ times exponentials can often be done by replacing $e^x$ by $e^{a x}$ and then noting that $d/da(e^{ax}) = x e^{ax}$, so we try to replace the powers of $x$ with derivatives of another expression. Then we can exploit this by bringing the derivative outside of the integral. For example
$$\int dx ~ x e^x = \left[ \frac{d}{da} \int dx~e^{ax} \right]_{a=1},$$
which you should be able to verify by doing both integrals explicitly.

In your case, we can use
$$ \frac{x^2 e^x}{(e^x+1)^2} = \left[ \frac{d^2}{da^2} \ln ( 1+ e^{ax})\right]_{a=1}.$$
Furthermore, we can determine the indefinite integral
$$ \int dx \ln ( 1+ e^{ax}) $$
in terms of the dilogarithm function (see for instance https://en.wikipedia.org/wiki/Spence's_function)
$$\text{Li}_2(z) = - \int^z_0 \frac{du}{u} \ln ( 1-u).$$

The big difficulty here is that the dilogarithm is infinite as $z\rightarrow -\infty$, so the naive substitution for your integral over the whole real axis will result in a divergent integral. (The dilogarithm is also usually not defined for $1 \leq z < \infty$, but I believe that the proper substitutions keep us on the negative real axis.) However, I believe that it is possible to show that the definite integral
$$ F(a) =\int_{-\infty}^0 dx \ln ( 1+ e^{ax}) $$
exists. So we should break your original integral into two parts, then the answer can be expressed as the appropriate derivative of $F(a)+F(-a)$.

It will probably be important to use the results (https://en.wikipedia.org/wiki/Spence's_function#Special_values) $\text{Li}_2(-1)=-\pi^2/12$ and $\text{Li}_2(0)=0.$

 

[Tphysics]

Thanks but this is math I am completely unfamiliar with. It ended up being doable also with a contour integral.

SOLVED.

 

[Tphysics]

I drew it terribly above but you catch my drift.

 

[fzero]

I drew it terribly above but you catch my drift.

Sure, I didn't seriously consider suggesting the contour integral because it is a bit rare to find someone comfortable with the method. I probably should have asked first. It's good that you were able to work it out yourself that way.