How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

Question

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$

I tried applying residuals, but the pole is of infinite(?) order.

Answer 1

Here, we present an approach that uses "Feynmann's Trick" for differentiating under the integral along with Contour Integration.

Let $I$ be the integral given by

$$I=\int_{-\infty}^\infty \frac{x^2e^x}{(e^x+1)^2}\,dx$$

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"FEYNMANN'S TRICK"

Enforcing the substitution $x\to \log(x)$ reveals

$$\begin{align} I&=\int_0^\infty \frac{\log^2(x)}{(x+1)^2}\,dx\\\\ &=\bbox[5px,border:2px solid #C0A000]{\left.\left(\frac{d^2}{da^2}\int_0^\infty \frac{x^a}{(x+1)^2}\,dx\right)\right|_{a=0}} \tag 1 \end{align}$$

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CONTOUR INTEGRATION

To evaluate the integral in $(1)$, we move to the complex plane and analyze the closed-contour integral $J(a)$ given by

$$\bbox[5px,border:2px solid #C0A000]{J(a)=\oint_C \frac{z^a}{(1+z)^2}\,dz} \tag 2$$

where $C$ is the classical "key-hole" contour along the branch cut extending from the origin along the non-negative real axis.

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Evaluation Using the Residue Theorem

From the Residue Theorem, $J(a)$ is given by

$$\begin{align} J(a)&=2\pi i \text{Res}\left(\frac{z^a}{(1+z)^2}, z=-1\right)\\\\ &=2\pi i \left.\frac{d}{dz}\left((1+z)^2\frac{z^a}{(1+z)^2}\right)\right|{z=-1}\\\\ &=\bbox[5px,border:2px solid #C0A000]{-2\pi i a e^{i\pi a}} \tag 2 \end{align}$$

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Decomposing $J(a)$

Next, we write $J(a)$ as

$$\begin{align} J(a)&=\int_0^\infty \frac{x^a}{(1+x)^2}\,dx-\int_0^\infty \frac{x^ae^{i2\pi a}}{(1+x)^2}\,dx\\\\ &=\bbox[5px,border:2px solid #C0A000]{(1-e^{i2\pi a})\int_0^\infty \frac{x^a}{(1+x)^2}\,dx} \tag 3 \end{align}$$

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PUTTING THINGS TOGETHER

From $(2)$ and $(3)$ we see that

$$\bbox[5px,border:2px solid #C0A000]{\int_0^\infty \frac{x^a}{(1+x)^2}\,dx=\frac{\pi a}{\sin(\pi a)}} \tag 4$$

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FINISHING IT OFF

Finally, using $(4)$ in $(1)$ reveals

$$\begin{align} I&=\left.\left(\frac{d^2}{da^2}\frac{\pi a}{\sin(\pi a)}\right)\right|_{a=0}\\\\ &=\lim_{a\to 0}\left(\frac{\pi^3 a(1+\cos^2(\pi a))-2\pi^2 \cos(\pi a)\sin(\pi a)}{\sin^3(\pi a)}\right)\\\\\ &=\frac{\pi^2}{3} \end{align}$$

as was to be shown!

Answer 2

We have $$\frac d{dx}\frac1{e^x+1}=\frac{-e^x}{(e^x+1)^2}$$ Also $$\frac{e^x}{(e^x+1)^2}=\frac{e^{-x}}{(1+e^{-x})^2}$$ So $$\begin{align}\int_{-\infty}^{\infty}x^2\frac{e^x}{(e^x+1)^2}dx&=2\int_0^{\infty}x^2\frac{e^x}{(e^x+1)^2}dx=-2\int_0^{\infty}x^2\frac d{dx}\frac1{e^x+1}dx\\ &=\left.-2x^2\frac1{e^x+1}\right|_0^{\infty}+4\int_0^{\infty}\frac x{e^x+1}dx\\ &=0+4\int_0^{\infty}\frac{xe^{-x}}{e^{-x}+1}dx=4\sum_{k=0}^{\infty}(-1)^k\int_0^{\infty}xe^{-(k+1)x}dx\\ &=4\sum_{k=0}^{\infty}(-1)^k \frac{\Gamma(2)}{(k+1)^2}=4\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^2}\\ &=4\left(1-\frac24\right)\zeta(2)=2\frac{\pi^2}6=\frac{\pi^2}3\end{align}$$

Answer 3

Another approach. We have $$\int_{-\infty}^{\infty}\frac{x^{2}e^{x}}{\left(e^{x}+1\right)^{2}}dx=2\int_{0}^{\infty}\frac{x^{2}e^{x}}{\left(e^{x}+1\right)^{2}}dx=2\int_{0}^{\infty}\frac{x^{2}}{e^{x}+1}dx-2\int_{0}^{\infty}\frac{x^{2}}{\left(e^{x}+1\right)^{2}}dx. $$ The first integral is classical $$2\int_{0}^{\infty}\frac{x^{2}}{e^{x}+1}dx=2\sum_{k\geq0}\left(-1\right)^{k}\int_{0}^{\infty}x^{2}e^{-x\left(k+1\right)}dx $$ $$=4\sum_{k\geq0}\frac{\left(-1\right)^{k}}{\left(k+1\right)^{3}}=3\zeta\left(3\right) $$ and in a similar way we can compute the other integral $$2\int_{0}^{\infty}\frac{x^{2}}{\left(e^{x}+1\right)^{2}}dx=2\sum_{k\geq0}k\left(-1\right)^{k-1}\int_{0}^{\infty}x^{2}e^{-x\left(k+1\right)}dx $$ $$=4\sum_{k\geq0}\frac{k\left(-1\right)^{k-1}}{\left(k+1\right)^{3}}=-\frac{\pi^{2}}{3}+3\zeta\left(3\right) $$ hence $$\int_{-\infty}^{\infty}\frac{x^{2}e^{x}}{\left(e^{x}+1\right)^{2}}dx=\frac{\pi^{2}}{3}$$ as wanted.

Answer 4

I would start by trying to find the indefinite integral, as follows: $$ \frac{d}{dx} \frac{x2e^x}{(1+e^x)^2} = \frac{x^2e^x}{(1+e^x)^2}+2\frac{x e^x}{1+e^x} $$ Now let's try to rid ourselves of the $2\frac{x e^x}{1+e^x}$: $$ \frac{d}{dx} \left( 2x \log(1+e^x) \right) = 2\frac{x e^x}{1+e^x} + 2 \log(1+e^x) $$ Well, we now have to add back $2\int \log(1+e^x) dx$. But this is progress because dilogarithms are your friend. (You can tell I don't have many human friends.) In particular, $$ \frac{d}{du} \mbox{Li}_2 (u) = -\frac{\log(1-u)}{u} $$ and if we substitute $u=-e^x$ we can find that $$ \frac{d}{dx} \mbox{Li}_2 (-e^{x}) = -\log(1+e^x) $$ So we can find the indefinite integral: $$ \int\frac{x^2e^x}{(1+e^x)^2}dx = x^2\frac{e^x}{(1+e^x)^2}-2x\log(1+e^x) - 2\mbox{ Li}_2(-e^x) $$ After all of that, we deserve to be close to finished, but now we need to find the value of the improper definite integral. We want $$ \lim_{x\to\infty} \left[ x^2\frac{e^x}{(1+e^x)^2}-2x\log(1+e^x) - 2\mbox{ Li}_2(-e^x) -x^2\frac{e^{-x}}{(1+e^{_x})^2}-2x\log(1+e^{-x})+2\mbox{ Li}_2(-e^{-x})\right] $$ Of these six terms, the last three (that is, all three terms form the negative side of the improper definite integral) go to zero in the limit as $x->\infty$. We are left with $$ \lim_{x\to\infty} \left[ x^2\frac{e^x}{(1+e^x)^2}-2x\log(1+e^x) - 2\mbox{ Li}_2(-e^x) \right] $$ We know (at least if we have been on a couple of dates with dilogarithms) that $$ \lim_{x\to\infty} \mbox{ Li}_2(-e^x) +\frac{x^2}{2} = -\frac{\pi^2}{6} $$ It is easy to see that $$\lim_{x\to\infty} \left(x^2\frac{e^x}{(1+e^x)^2} -x^2 \right)=0 $$ And it is not too hard to see that $$\lim_{x\to\infty} \left( x\log(1+e^x) -x^2 \right)=0$$ (effectively, for large $x$, the $1$ in the log doesn't matter so the log looks like $x$).

Combining these limits, we find that our answer is $$ \lim_{x\to\infty} \left[ \left(x^2\frac{e^x}{(1+e^x)^2} -x^2\right)\\ -2\left(x\log(1+e^x) -x^2\right)\\- 2 \left(\mbox{ Li}_2(-e^x) +\frac12 x^2 \right)\\+x^2-2x^2+x^2\right]\\ = 0 -2 \cdot 0 -2 \cdot \left( -\frac{\pi^2}{6}\right) = \frac{\pi^2}{3} $$

Answer 5

Consider \begin{align} I(a)&=\int_0^\infty\frac{x}{1+e^{ax}}\ \mathrm dx\\[9pt] &=\int_0^\infty\frac{x\ e^{-ax}}{1+e^{-ax}}\ \mathrm dx\\[9pt] &=\int_0^\infty\sum_{k=0}^\infty\ (-1)^k\ xe^{-a(k+1)x}\ \mathrm dx\\[9pt] &=\sum_{k=0}^\infty\ (-1)^k\int_0^\infty xe^{-a(k+1)x}\ \mathrm dx\\[9pt] &=\sum_{k=0}^\infty\ \frac{(-1)^k}{a^2(k+1)^2}\\[9pt] &=\frac{\pi^2}{12a^2}\\[9pt] \end{align} then the given integral is simply $2I'(-1)$.