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A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.

%I A102886 #58 Feb 16 2025 08:32:56
%S A102886 2,7,2,1,9,8,2,6,1,2,8,7,9,5,0,2,6,6,3,1,2,5,8,6,1,1,2,2,7,9,7,0,1,7,
%T A102886 4,3,4,1,7,3,2,2,9,6,2,5,4,6,1,6,0,7,8,6,7,9,0,7,2,4,4,0,6,6,4,9,2,8,
%U A102886 8,5,6,8,6,4,7,0,9,2,7,4,8,3,0,3,7,9,1,1,2,0,2,0,1,3,3,2,8,7,8,1,3,2
%N A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx.
%C A102886 Named after the French mathematician Joseph-Alfred Serret (1819-1885). - _Amiram Eldar_, May 30 2021
%D A102886 Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264.
%D A102886 L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18.
%D A102886 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8.
%H A102886 Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.4)
%H A102886 J.-A. Serret, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k16388p/f442n1.capture">Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx</a>, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436.
%H A102886 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See pp. 308-309.
%H A102886 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SerretsIntegral.html">Serret's Integral</a>.
%F A102886 Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - _Jean-François Alcover_, Mar 25 2013
%F A102886 Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - _Bernard Schott_, Apr 23 2020
%F A102886 Equals Pi*log(2)/8 = Sum_{n>0} (-1)^(n+1) * H(2n) / (2n+1) = H(2)/3 - H(4)/5 + H(6)/7 -... with H(n) = Sum_{j=1..n} 1/j the harmonic numbers. [Jolley]; improved by _Bernard Schott_, Apr 24 2020
%F A102886 Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - _Amiram Eldar_, May 30 2021
%F A102886 Equals Integral_{x=0..log(2)} x/(e^x + 2*e^(-x) - 2) dx = -Integral_{x=0..Pi/2} log(sin(x))*sin(x)/sqrt(1+sin(x)^2) dx = Integral_{x=0..1} log((1 - x)/x)/(1 + x^2) dx = Integral_{x=0..Pi/4} x/((cos(x) + sin(x))*cos(x)) dx = Integral_{x=0..Pi/4} log(cot(x) - 1) dx (see Shamos). - _Stefano Spezia_, Nov 13 2024
%e A102886 0.27219826128795026631258611227970174341732296254616...
%t A102886 RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* _Jean-François Alcover_, May 17 2013 *)
%o A102886 (PARI) Pi*log(2)/8 \\ _Michel Marcus_, Apr 23 2020
%o A102886 (PARI) intnum(x=0, 1, log(x+1)/(x^2+1)) \\ _Michel Marcus_, Apr 26 2020
%Y A102886 Cf. A086054 (Pi*log(2)).
%K A102886 nonn,cons,easy
%O A102886 0,1
%A A102886 _Eric W. Weisstein_, Jan 15 2005