Definite Integral: $\int_0^1\frac{\ln^4(x)}{x^2+1}\,dx$

Question

I'm trying to derive a closed-form expression for

$$I=\int_0^1\frac{\ln^4(x)}{x^2+1}\,dx$$

Letting $u=-\ln(x), x=e^{-u}, dx=-e^{-u}\,du $ yields

$$I=\int_0^{\infty}\frac{u^4e^{-u}}{e^{-2u}+1}\,du$$

Setting $u\to-u$ and manipulating the integrands yield

$$I=-\int_0^{-\infty}\frac{u^4e^{u}}{e^{2u}+1}\,du$$ $$=\int_{-\infty}^0\frac{u^4e^{-u}}{e^{-2u}+1}\,du$$

And adding the two equivalent forms of $I$ yields

$$2I=\int_{-\infty}^{\infty}\frac{u^4e^{-u}}{e^{-2u}+1}\,du$$

I've tried to differentiate under the integral sign, but I could not find any parameterization that worked for me. (Perhaps someone could tell me how to solve such integrals by differentiation under the integral sign?)

My best attempt so far was using complex analysis:

I used a counterclockwise semicircle that grows to infinity over the lower half of the complex plane as my contour, and by Jordan's lemma (as I understand it) the integral over the arc vanishes and so I should be left with

$$\require{cancel} \lim_{R\to\infty} \int_R^{-R} \frac{x^4e^{-x}}{e^{-2x}+1}\,dx + \cancel{\int_{arc} \frac{z^4e^{-z}}{e^{-2z}+1}\,dz} = 2\pi i\sum_j \operatorname{Res}(j)$$

$$-2I=\int_{\infty}^{-\infty}\frac{x^4e^{-x}}{e^{-2x}+1}\,dx= 2\pi i\sum_j \operatorname{Res}(j)$$

Since my integrand only blows up when $e^{-2u}+1=0 \Rightarrow u=-i\pi/2$,

$$\frac{-2}{2\pi i}I=\operatorname{Res}(-i\pi/2)$$

$$\frac{i}{\pi} I = \lim_{z\to -i\pi/2}(z+i\pi/2)\frac{z^4e^{-z}}{e^{-2z}+1}$$

Evaluating the limit (via L'Hopital's Rule and a few substitutions) yields

$$\frac{i}{\pi}I = \frac{i\pi^4}{32}$$

$$I=\frac{\pi^5}{32}$$

However, WolframAlpha evaluates the integral at $$I=\frac{5\pi^5}{64}$$

Where did I make a mistake and how do I evaluate this integral correctly?

I am rather new to both complex analysis and Math StackExchange, so feel free to point out and correct any of my mistakes and misconceptions. Any help is greatly appreciated!

Answer 1

You got to

$I=\int_0^\infty u^4e^{-u}(1+e^{-2u})^{-1}du$

which gives you

$I=\int_0^\infty u^4e^{-u}(1-e^{-2u}+e^{-4u}-e^{-6u}+...)du$

or

$I=\int_0^\infty u^4(e^{-u}-e^{-3u}+e^{-5u}-e^{-7u}+...)du$

From standard integration tables or integration by parts you should be able to get your answer ...

Answer 2

An approach relying on Feynman's trick

Notice that one has: \begin{align} I:=\int^1_0 \frac{\ln^4(x)}{x^2+1}\,dx \stackrel{x\mapsto 1/x}{=} \int^\infty_1 \frac{\ln^4(x)}{x^2+1}\,dx \end{align} which means that: \begin{align} I = \frac 1 2 \int^\infty_0 \frac{\ln^4(x)}{x^2+1}\,dx \end{align} Define: \begin{align} G(z):=\int^\infty_0 \frac{x^{-z}}{x^2+1}\,dx \end{align} Notice by Feynman's trick one has: \begin{align} \frac 1 2 G^{(4)}(0) =I \end{align} So we only need to find $G(z)$ which is not very hard. You can see for example this post for a variety of solutions. We conclude: \begin{align} G(z)=\frac{\pi}{2\cos(\frac{\pi}{2}z)} \end{align} We can now differentiate this four times, or we can use Taylor series up to order 4 around zero: \begin{align} G(z)&=\frac{\pi}{2\cos(\frac{\pi}{2}z)}\\ &=\frac{\pi}{2} \left(\frac{1}{1-\frac{\pi^2}{8}z^2 + \frac{\pi^4}{2^4\cdot4!}z^4+O(z^5)}\right)\\ &=\frac{\pi}{2}\left[ 1+\left(\frac{\pi^2}{8}z^2 - \frac{\pi^4}{2^4\cdot 4!}z^4+O(z^5) \right) + \left(\frac{\pi^2}{8}z^2 - \frac{\pi^4}{2^4\cdot 4!}z^4+O(z^5) \right)^2+O(z^5)\right]\\ &=\frac{\pi}{2}+\frac{\pi^3}{16}z^2+\frac{5\pi^5}{32\cdot 4! }z^4+O(z^5)\\ \end{align} The coefficient of $z^4$ gives $4!G^{(4)}(0)$ hence: \begin{align} G^{(4)}(0) = \frac{\pi^55}{32 } \end{align} We conclude: \begin{align} I=\frac{5\pi^5 }{64} \end{align}

Answer 3

A Complex-Analytic Proof

Your problem is that the integral does not vanish on the semicircular arc as the radius goes to infinity, and there are infinitely many poles that your contour will end up enclosing. I am offering a similar approach, but with a different contour that guarantees that it encloses only one pole, and that the un-needed terms vanish as the range expands.

Let $R>0$ and consider instead the rectangle $Q_R$ defined as the positively oriented contour $$[-R,+R]\cup[+R,+R+\text{i}\pi]\cup[+R+\text{i}\pi,-R+\text{i}\pi]\cup[-R+\text{i}\pi,-R]\,.$$ Define $$L_k:=\lim_{R\to\infty}\,\oint_{Q_R}\,f_k(z)\,\text{d}z\,,\text{ where }f_k(z):=z^k\,\left(\frac{\exp(z)}{\exp(2z)+1}\right)\,.$$ We shall attempt to determine the values of $L_k$ for $k=0,2,4$. Using the Residue Theorem, it is easy to see that $$L_k=2\pi\text{i}\,\text{Res}_{z=\frac{\text{i}\pi}{2}}\big(f_k(z)\big)\,,$$ so that $$L_0=\pi\,,\,\,L_2=-\frac{\pi^3}{4}\,,\text{ and }L_4=\frac{\pi^5}{16}\,.$$

Write $$J_k:=\int_{-\infty}^{+\infty}\,f_k(u)\,\text{d}u\,.$$ It is not difficult to show that $$L_0=2\,J_0\,,\,\,L_2=2\,J_2-\pi^2\,J_0\,,\text{ and }L_4=2\,J_4-6\pi^2\,J_2+\pi^4\,J_0\,.$$ Thus, we get $$J_0=\frac{\pi}{2}\,,\,\,J_2=-\frac{\pi^3}{8}+\frac{\pi^3}{4}=\frac{\pi^3}{8}\,,$$ and $$J_4=\frac{\pi^5}{32}+\frac{3\pi^5}{8}-\frac{\pi^5}{4}=\frac{5\pi^5}{32}\,.$$ Thus, $$I=\frac{1}{2}\,J_4=\frac{5\pi^5}{64}\,.$$ You can obtain $$I_k:=\int_0^\infty\,u^k\,\left(\frac{\exp(u)}{\exp(2u)+1}\right)\,\text{d}u$$ similarly for an even integer $k\geq 0$, by evaluating $L_0,L_2,L_4,\ldots,L_k$ and then solving for $J_0,J_2,J_4,\ldots,J_k$, as $I_k=\dfrac{1}{2}\,J_k$. From here, we can show that $$J_k=t_k\,\left(\frac{\pi^{k+1}}{2^{k+1}}\right)\text{ for all even integers }k\geq 0\,,$$ where $$t_k=\sum_{r=0}^{\frac{k}{2}-1}\,(-1)^r\,\binom{k}{2r+2}\,2^{2r+1}\,t_{k-2r-2}+(-1)^{\frac{k}{2}}\,.$$ For example, $t_0=1$, $t_2=1$, $t_4=5$, $t_6=61$, and $t_8=1385$.

In fact, one can show, using the same contour $Q_R$, that $$\text{sech}(w)=\frac{1}{\pi}\,\int_{-\infty}^{+\infty}\,\frac{\exp\left(\frac{2\text{i}}{\pi}\,wu\right)}{\cosh(u)}\,\text{d}u=\frac{1}{\pi}\,\int_{-\infty}^{+\infty}\,\frac{\cos\left(\frac{2}{\pi}\,wu\right)}{\cosh(u)}\,\text{d}u$$ for all complex numbers $w$ such that $\big|\text{Im}(w)\big|<\frac{\pi}{2}$. This shows that $t_k=(-1)^{\frac{k}{2}}\,E_k=|E_k|$ for every even integer $k\geq 0$, where $E_0,E_1,E_2,\ldots$ are Euler numbers. Therefore, $$\sum_{n=0}^\infty\,\frac{(-1)^n\,k!}{(2n+1)^{k+1}}={\small\int_0^\infty\,u^k\,\left(\frac{\exp(u)}{\exp(2u)+1}\right)\,\text{d}u}=I_k=\frac{1}{2}\,J_k=\frac{t_k}{2}\,\left(\frac{\pi}{2}\right)^{k+1}=\frac{|E_k|}{2}\,\left(\frac{\pi}{2}\right)^{k+1}$$ for each even integer $k\geq 0$.

In general, for $p\in\mathbb{C}$ and $q\in\mathbb{R}_{>0}$ such that $0<\text{Re}\left(p\right)<q$, we have $$\begin{align}\int_{-\infty}^{+\infty}\,\frac{u^k\,\exp(pu)}{\exp(qu)+1}\,\text{d}u&=k!\,\left(\frac{\pi}{q}\right)^{k+1}\,\Biggl(\left[a^k\right]\Bigg(\text{csc}\left(a+\frac{\pi p}{q}\right)\Bigg)\Biggr) \\ &=k!\,\left(\frac{\pi}{q}\right)^{k+1}\,\Biggl(\left[a^k\right]\Bigg(\text{sec}\left(a+\frac{\pi(2p-q)}{2q}\right)\Bigg)\Biggr)\,.\end{align}$$ Here, $\left[a^k\right]\big(g(a)\big)$ is the coefficient of $a^k$ in the Laurent expansion of $g(a)$ about $a=0$.

Answer 4

I believe that the most simple approach is just to exploit Maclaurin series. Since $\int_{0}^{1}x^{2n}\log^4(x)\,dx=\frac{24}{(2n+1)^5}$ we have

$$ \int_{0}^{1}\frac{\log^4(x)}{x^2+1}\,dx = 24\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^5}\color{red}{=}24\cdot\frac{5\pi^5}{1536} = \frac{5\pi^5}{64}. $$ It is well-known that the series $\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^{2m+1}}$ are related to Euler numbers.

Answer 5

Let, for $n\geq 0$ integer,

$\begin{align}&A_n=\int_0^1 \frac{\ln^{2n}x}{1+x^2}\,dx\\ &B_n=\int_0^\infty \frac{\ln^{2n}x}{1+x^2}\,dx\\ &K_n=\int_0^\infty\int_0^\infty\frac{\ln^{2n}(xy)}{(1+x^2)(1+y^2}\,dx\,dy \end{align}$

Observe that,

$\begin{align}A_0&=\int_0^1 \frac{1}{1+x^2}\,dx\\ &=\Big[\arctan x\Big]_0^1\\ &=\frac{\pi}{4}\end{align}$

$\begin{align}B_n=\int_0^1 \frac{\ln^{2n}x}{1+x^2}\,dx+\int_1^\infty \frac{\ln^{2n}x}{1+x^2}\,dx\\ \end{align}$

Perform in the latter integral the change of variable $y=\dfrac{1}{x}$,

$\begin{align}B_n=2A_n\end{align}$

That is,

$\begin{align}A_n=\frac{1}{2}B_n\end{align}$

For $n\geq 0$ integer,

$\begin{align}\int_0^\infty \frac{\ln^{2n+1}x}{1+x^2}\,dx=0\end{align}$

(perform the change of variable $y=\dfrac{1}{x}$, and, $z=-z \iff z=0$ )

$\begin{align}K_n&=\int_0^\infty\int_0^\infty\left(\sum_{k=0}^{2n}\binom{2n}{k}\frac{\ln^k x\ln^{2n-k}y}{(1+x^2)(1+y^2)} \right)\,dx\,dy\\ &=\sum_{k=0}^{2n}\binom{2n}{k}\left(\int_0^\infty\frac{\ln^k x}{1+x^2}\,dx\right)\left(\int_0^\infty\frac{\ln^{2n-k}y}{1+y^2} \,dy\right)\\ &=\sum_{k=0}^{n}\binom{2n}{2k}\left(\int_0^\infty\frac{\ln^{2k} x}{1+x^2}\,dx\right)\left(\int_0^\infty\frac{\ln^{2(n-k)}y}{1+y^2} \,dy\right)\\ &=\sum_{k=0}^{n}\binom{2n}{2k}B_kB_{n-k} \end{align}$

Perform the change of variable $u=xy$,

$\begin{align}K_n&=\int_0^\infty\int_0^\infty\frac{\ln^{2n}(xy)}{(1+x^2)(1+y^2)}\,dx\,dy\\ &=\int_0^\infty\int_0^\infty\frac{y\ln^{2n}(u)}{(y^2+u^2)(1+y^2)}\,du\,dy\\ &=\int_0^\infty\int_0^\infty\frac{\ln^{2n}(u)}{u^2-1}\left(\frac{y}{1+y^2}-\frac{y}{u^2+y^2}\right)\,du\,dy\\ &=\frac{1}{2}\int_0^\infty\int_0^\infty\frac{\ln^{2n}(u)}{u^2-1}\left[\frac{1+y^2}{u^2+y^2}\right]_{y=0}^{y=\infty}\,du\\ &=\int_0^\infty\frac{\ln^{2n+1}(u)}{u^2-1}\,du\\ &=\int_0^1\frac{\ln^{2n+1}(u)}{u^2-1}\,du+\int_1^\infty\frac{\ln^{2n+1}(u)}{u^2-1}\,du \end{align}$

In the latter integral perform the change of variable $x=\dfrac{1}{u}$,

$\begin{align}K_n&=2\int_0^1\frac{\ln^{2n+1}(x)}{x^2-1}\,dx\\ &=2\int_0^1\frac{\ln^{2n+1}(x)}{x-1}\,dx-2\int_0^1\frac{x\ln^{2n+1}(x)}{x^2-1}\,dx\\ \end{align}$

In the latter integral perform the change of variable $u=x^2$,

$\begin{align}K_n&=2\int_0^1\frac{\ln^{2n+1}(x)}{x-1}\,dx-\frac{1}{2^{2n+1}}\int_0^1\frac{\ln^{2n}(u)}{u-1}\,du\\ &=\left(2-\frac{1}{2^{2n+1}}\right)\int_0^1\frac{\ln^{2n+1}(x)}{x-1}\,dx\\ &=-\left(2-\frac{1}{2^{2n+1}}\right)\int_0^1 \ln^{2n+1}(x)\left(\sum_{k=0}^\infty x^k\right)\,dx\\ &=-\left(2-\frac{1}{2^{2n+1}}\right)\sum_{k=0}^\infty \left(\int_0^1 x^k \ln^{2n+1}(x)\,dx\right)\\ &=-\left(2-\frac{1}{2^{2n+1}}\right)\sum_{k=0}^\infty \frac{(-1)^{2n+1}(2n+1)!}{(k+1)^{2n+2}}\\ &=\left(2-\frac{1}{2^{2n+1}}\right)(2n+1)!\zeta(2n+2) \end{align}$

Therefore,

$\begin{align}\sum_{k=0}^{n}\binom{2n}{2k}B_kB_{n-k}&=\left(2-\frac{1}{2^{2n+1}}\right)(2n+1)!\zeta(2n+2)\end{align}$

Therefore,

$\begin{align}\boxed{\sum_{k=0}^{n}\binom{2n}{2k}A_kA_{n-k}=\frac{(2n+1)!}{2}\left(1-\frac{1}{2^{2n+2}}\right)\zeta(2n+2)}\end{align}$

if $n=1$,

$\begin{align} \frac{\pi}{2}A_1=\frac{45}{16}\zeta(4)\end{align}$

if $n=2$,

$\begin{align} \frac{\pi}{2}A_2+6A_1^2=\frac{945}{16}\zeta(6)\end{align}$

therefore,

$\begin{align} \boxed{A_2=\frac{945\zeta(6)}{8\pi}-\frac{6075\zeta(4)^2}{16\pi^3}}\end{align}$

If you know that,

$\begin{align}&\zeta(4)=\frac{1}{90}\pi^4\\ &\zeta(6)=\frac{1}{945}\pi^6 \end{align}$

then,

$\begin{align}\boxed{A_2=\frac{5}{64}\pi^5}\end{align}$

Answer 6

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} I & \equiv \int_{0}^{1}{\ln^{4}\pars{x} \over x^{2} + 1}\,\dd x = \left.\totald[4]{}{\nu}\int_{0}^{1}{x^{\nu} \over 1 + x^{2}}\,\dd x \,\right\vert_{\ \nu\ =\ 0} = \left.\totald[4]{}{\nu}\int_{0}^{1}{x^{\nu} - x^{\nu + 2}\over 1 - x^{4}}\,\dd x \,\right\vert_{\ \nu\ =\ 0} \\[5mm] & \stackrel{x^{4}\ \mapsto\ x}{=}\,\,\, \left.{1 \over 4}\,\totald[4]{}{\nu}\int_{0}^{1}{x^{\nu/4 - 3/4} - x^{\nu/4 - 1/4}\over 1 - x}\,\dd x \,\right\vert_{\ \nu\ =\ 0} \\[5mm] & = {1 \over 4}\,\totald[4]{}{\nu}\pars{% \int_{0}^{1}{1 - x^{\nu/4 - 1/4}\over 1 - x}\,\dd x - \int_{0}^{1}{1 - x^{\nu/4 - 3/4}\over 1 - x}\,\dd x}_{\ \nu\ =\ 0} \end{align}

With $\mathbf{\color{black}{6.3.22}}$ A & S identity:

\begin{align} I & \equiv \int_{0}^{1}{\ln^{4}\pars{x} \over x^{2} + 1}\,\dd x \\[5mm] & = {1 \over 4}\,\totald[4]{}{\nu}\bracks{\Psi\pars{{\nu \over 4} + {3 \over 4}} - \Psi\pars{{\nu \over 4} + {1 \over 4}}}\qquad \pars{~\Psi:\ Digamma\ Function~} \\[5mm] & = {1 \over 4}\pars{1 \over 4}^{4}\bracks{\Psi^{\pars{\texttt{IV}}}\pars{3 \over 4} - \Psi^{\pars{\texttt{IV}}}\pars{1 \over 4}} \\[5mm] & = {1 \over 1024}\, \left.\totald[4]{\bracks{\pi\cot\pars{\pi z}}}{z}\,\right\vert_{\ z\ =\ 1/4} \qquad\pars{~Euler\ Reflection\ Formula} \\[5mm] & = {1 \over 1024}\ \underbrace{\bracks{8\pi^{5}\cot^{3}\pars{\pi z}\csc^{2}\pars{\pi z} + 16\pi^{5}\cot\pars{\pi z}\csc^{4}\pars{\pi z}}_{\ z\ =\ 1/4}} _{\ds{80\pi^{5}}} \\[5mm] & = \bbx{5\pi^{5} \over 64} \approx 23.9078 \end{align}