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

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, 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, 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

Offset

0,1

Comments

Named after the French mathematician Joseph-Alfred Serret (1819-1885). - Amiram Eldar, May 30 2021

References

Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264.

L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8.

Links

Table of n, a(n) for n=0..101.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.4)

J.-A. Serret, Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436.

Eric Weisstein's World of Mathematics, Serret's Integral.

Formula

Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - Jean-François Alcover, Mar 25 2013

Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - Bernard Schott, Apr 23 2020

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

Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - Amiram Eldar, May 30 2021

Example

0.27219826128795026631258611227970174341732296254616...

Mathematica

RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* Jean-François Alcover, May 17 2013 *)

Prog

(PARI) Pi*log(2)/8 \\ Michel Marcus, Apr 23 2020
(PARI) intnum(x=0, 1, log(x+1)/(x^2+1)) \\ Michel Marcus, Apr 26 2020

Crossrefs

Cf. A086054 (Pi*log(2)).

Sequence in context: A095194 A254251 A095711 * A204382 A072981 A023399

Adjacent sequences: A102883 A102884 A102885 * A102887 A102888 A102889

Keyword

nonn,cons,easy

Author

Eric W. Weisstein, Jan 15 2005

Status

approved