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Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from the definite aspect only? Wolfram Alpha suggests that the indefinite version is miserable.

Thank you!

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    $\begingroup$ Is contour integration allowed? Or only real methods? $\endgroup$ – dustin Dec 20 '14 at 18:27
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    $\begingroup$ This integral has been done many times on this site. Taylor expand the denominator, use the fact that $$\int_0^1 dx \, x^{2 k} \log{x} = -\frac1{(2 k+1)^2}$$ and sum. $\endgroup$ – Ron Gordon Dec 20 '14 at 18:29
  • $\begingroup$ I don't think it's needed. $\endgroup$ – Spinning Turntable Dec 20 '14 at 18:29
  • $\begingroup$ this integral gives the Catalan constant $\endgroup$ – Dr. Sonnhard Graubner Dec 20 '14 at 18:41
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Use integration by parts to get $$ \displaystyle\int_0^1 \frac {\ln x}{1 + x^2} \, \mathrm{d}x = \left[ \arctan(x) \ln(x) \right]_0^1 - \displaystyle\int_0^1 \frac {\arctan(x)}{x} \, \mathrm{d}x = - \displaystyle\int_0^1 \frac {\arctan(x)}{x} \, \mathrm{d}x. $$Now, we use the Taylor expansion of $\arctan(x)$ to get $$ -\int_0^1 \sum_{n=0}^{\infty}\frac{1}{x}\frac{(-1)^nx^{2n+1}}{2n+1}\,\mathrm{d}x $$Switch the sum and the integral and it should be easy to end at the result $ -\displaystyle\sum_{n=0}^{\infty} \frac {(-1)^n}{(2n+1)^2} $.

This is now the Catalan constant.

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    $\begingroup$ What is "the $n$th Catalan constant"? $\endgroup$ – Sasha Dec 20 '14 at 18:37
  • $\begingroup$ @Sasha Oops, typo. Fixed, thanks! $\endgroup$ – Ahaan S. Rungta Dec 20 '14 at 18:38
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let $-I = \int_0^1{\ln x \over 1 + x^2} \ dx.$ we will need $$a_k = \int_0^\infty xe^{-kx}\ dx = {1 \over k^2} \int_0^\infty xe^{-x} \ dx = {1 \over k^2}$$

$I = \int_0^1{\ln x \over 1 + x^2} \ dx,$ by a change of variable $ u = -\ln x, x = e^{-u}, $ the integral

$\begin{eqnarray} I &=& \int_0^\infty{ ue^{-u} \over 1 + e^{-2u}} \ du \\ &=& \int_0^\infty ue^{-u}\left(1 - e^{-2u} + e^{-4u}+ \cdots \right) du\\ &=& a_1 - a_3 + a_5 + \cdots \\ &=&1 - {1 \over 3^2} + {1 \over 5^2} -{1 \over 7^2}+ \cdots = 0.9159\cdots= \mbox{ Catalan constant} \end{eqnarray}$

p.s.: i just learned the name for this constant.

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\begin{align} \int_{0}^{1}\frac{\ln x}{1+x^2}\mathrm dx&=\int_0^{1}\sum_{n=1}^\infty(-1)^n\, x^{2n}\ln x\;\mathrm dx\tag1\\ &=\sum_{n=1}^\infty(-1)^n\int_0^{1} x^{2n}\ln x\;\mathrm dx\tag2\\ &=-\sum_{n=1}^\infty\frac{(-1)^n}{(2n+1)^2}\tag3\\ &=\bbox[5pt,border:3px #FF69B4 solid]{\color{red}{\large-\text{G}}} \end{align}


Explanation :

$(1)\;$ Use series expansion $\;\displaystyle \frac{1}{1+y}=\sum_{n=1}^\infty(-1)^n \,y^{n}$ and replace $y=x^2$

$(2)\;$ Use formula $\;\displaystyle \int_0^1 x^n \ln^k x\ dx=\frac{(-1)^k\, k!}{(n+1)^{k+1}}$, for $\ k=0,1,2,\cdots$

$(3)\;$ Series expression of Catalan's constant

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$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\dsc}[1]{\displaystyle{\color{red}{#1}}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\Li}[1]{\,{\rm Li}_{#1}} \newcommand{\norm}[1]{\left\vert\left\vert\, #1\,\right\vert\right\vert} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ \begin{align}&\color{#66f}{\large\int_{0}^{1}{\ln\pars{x} \over 1 + x^{2}}\,\dd x} =\Re\int_{0}^{1}{\ln\pars{x} \over 1 - \ic x}\,\dd x =\Im\int_{0}^{1}{\ln\pars{-\ic\bracks{\ic x}} \over 1 - \ic x} \,\pars{\ic\,\dd x} =\Im\int_{0}^{\ic}{\ln\pars{-\ic x} \over 1 - x}\,\dd x \\[5mm]&=\Im\bracks{\left.\vphantom{\LARGE A} -\ln\pars{1 - x}\ln\pars{-\ic x}\right\vert_{\, 0}^{\,\ic} +\int_{0}^{\ic}{\ln\pars{1 - x} \over x}\,\dd x} =-\,\Im\int_{0}^{\ic}\Li{2}'\pars{x}\,\dd x =-\,\Im\Li{2}\pars{\ic} \\[5mm]&=-\,\Im\sum_{n\ =\ 1}^{\infty}{\ic^{n} \over n^{2}} =-\,\Im\sum_{n\ =\ 1}^{\infty}{\ic^{2n - 1} \over \pars{2n - 1}^{2}} =-\,\Im\bracks{% \ic^{-1}\sum_{n\ =\ 1}^{\infty}{\pars{-1}^{n} \over \pars{2n - 1}^{2}}} =-\sum_{n\ =\ 0}^{\infty}{\pars{-1}^{n} \over \pars{2n + 1}^{2}} \\[5mm]&=\color{#66f}{\Large-G} \end{align}

where $\ds{G}$ is the Catalan Constant.

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