A225113 - OEIS

%I #28 Nov 02 2024 05:39:12

%S 6,5,8,4,7,2,3,2,5,6,9,9,6,3,4,1,3,6,4,8,7,0,9,8,8,9,7,1,6,6,0,0,5,2,

%T 7,5,9,0,5,5,8,1,7,5,6,2,4,9,0,4,1,8,5,7,2,6,2,7,9,5,3,5,1,2,0,5,1,7,

%U 0,8,7,9,6,6,7,6,6,8,2,2,7,7,6,3,3,3,8,6,3,7,0,6,3,1,2,9,9,9,4,5

%N Decimal expansion of Integral_{x=0..1} arcsin(x)^2/x dx.

%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.

%H Chenli Li, Wenchang Chu, <a href="http://dx.doi.org/10.3390/math10162980">Improper integrals involving powers of Inverse Trigonometric and hyperbolic Functions</a>, Mathematics 10 (16) (2022) 2980

%H D. Cvijovic, H. M. Srivastava, <a href="https://dx.doi.org/10.1016/j.jmaa.2008.10.017">Evaluations of some classes of the trigonometric moment intgrals</a>, J. Math. Anal. Applic. 351 (2009) 244-256, example set 1.

%H J. C. Tanner, <a href="http://www.jstor.org/stable/3213813">The Proportion of Quadrilaterals Formed by Random Lines in a Plane</a>, Journal of Applied Probability, Vol. 20, No. 2 (Jun., 1983), pp. 400-404.

%H J. C. Tanner, <a href="http://www.jstor.org/stable/3213589">Polygons Formed by Random Lines in a Plane: Some Further Results</a>, Journal of Applied Probability, Vol. 20, No. 4 (Dec., 1983), p. 778

%F Equals 1/8*(Pi^2*log(4) - 7*zeta(3)).

%F Also equals Sum_{n>=1} 4^(n-1)/(n^3*binomial(2*n, n)).

%F Also equals 1/2*hypergeometric4F3([1, 1, 1, 1], [3/2, 2, 2], 1).

%F Also equals Integral_{x=0..Pi/2} x^2*cot(x) dx. - _Michel Marcus_, Aug 29 2015

%e 0.65847232569963413648709889716600527590558175624904185726279535120517...

%t 1/8*(Pi^2*Log[4] - 7*Zeta[3]) // RealDigits[#, 10, 100]& // First

%o (PARI) (Pi^2*log(4) - 7*zeta(3))/8 \\ _Michel Marcus_, Sep 04 2015

%Y Cf. A002117 (zeta(3)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 29 2013