Board index Computation of Integrals Two special logarithmic integral

Two special logarithmic integral

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Post Sun Oct 12, 2014 1:18 pm

Posts: 98
$$\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)L{i_3}\left( {\frac{{1 + x}}{2}} \right)}}{x}dx} = ?,\int\limits_0^1 {\frac{{{{\ln }^2}\left( {1 - x} \right)L{i_2}\left( {\frac{{1 + x}}{2}} \right)}}{x}dx} = ?.$$

Post Sun Oct 12, 2014 6:23 pm

Posts: 4

$\displaystyle{\large\int}_0^1\frac{\ln(1-x)\,\operatorname{Li}_3\left(\frac{1+x}2\right)}xdx=\frac{29\,\zeta(5)}{16}-\frac{19\pi^2}{96}\zeta(3)+\frac{5\,\zeta(3)}{16}\ln^22+\frac{\ln^52}{40}-\frac{5\pi^2}{72}\ln^32+\frac{11\pi^4}{1440}\ln2-3\operatorname{Li}_5\left(\tfrac12\right).$

$\displaystyle{\large\int}_0^1\frac{\ln^2(1-x)\,\operatorname{Li}_2\left(\frac{1+x}2\right)}xdx=\frac{81\,\zeta(5)}{32}+\frac{5\pi^2}{16}\zeta(3)-\frac{\zeta(3)}8\ln^22+\frac1{15}\ln^52-\frac{\pi^2}{18}\ln^32-\frac{\pi^4}{15}\ln2+2\operatorname{Li}_5\left(\tfrac12\right)+2\operatorname{Li}_4\left(\tfrac12\right)\ln2.$

Post Sun Oct 12, 2014 7:04 pm

Posts: 83
Incredible!

Post Sun Oct 12, 2014 11:20 pm

Posts: 98
CleoMSE wrote:
$\displaystyle{\large\int}_0^1\frac{\ln(1-x)\,\operatorname{Li}_3\left(\frac{1+x}2\right)}xdx=\frac{29\,\zeta(5)}{16}-\frac{19\pi^2}{96}\zeta(3)+\frac{5\,\zeta(3)}{16}\ln^22+\frac{\ln^52}{40}-\frac{5\pi^2}{72}\ln^32+\frac{11\pi^4}{1440}\ln2-3\operatorname{Li}_5\left(\tfrac12\right).$

$\displaystyle{\large\int}_0^1\frac{\ln^2(1-x)\,\operatorname{Li}_2\left(\frac{1+x}2\right)}xdx=\frac{81\,\zeta(5)}{32}+\frac{5\pi^2}{16}\zeta(3)-\frac{\zeta(3)}8\ln^22+\frac1{15}\ln^52-\frac{\pi^2}{18}\ln^32-\frac{\pi^4}{15}\ln2+2\operatorname{Li}_5\left(\tfrac12\right)+2\operatorname{Li}_4\left(\tfrac12\right)\ln2.$

@CleoMSE.Do you have specific process?

Post Mon Oct 13, 2014 12:55 am

Posts: 24
If she were to tell you how she got that, she wouldn't be Cleo...

(She frequents Math Stack Exchange, and always gives her answers without telling how she got them.)

Post Sun Sep 11, 2016 4:41 am

Posts: 1
This is strange for anyone how the process of thinking is in her (cleo)mind


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