A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000, The Putnam Archive. See Problem A-4.

Editors of the Mathematics Magazine, The 61st Annual William Lowell Putnam Examination, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.

Jihyung Kang and others, Convergence of I = Integral_0^oo sinx sin(x2) dx, Mathematics StackExchange, 2017.

Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-first William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.

Eric Weisstein's World of Mathematics, Fresnel Integrals.

Wikipedia, Fresnel integral.

Formula

Equals sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.

Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.

Example

0.49169967769382111771654625416890810022151027126875...

Mathematica

RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
(* Alternative: *)
RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]

Crossrefs

Cf. A069998, A217481, A231863.

Sequence in context: A388491 A388962 A388669 * A120865 A219031 A243452

Adjacent sequences: A387210 A387211 A387212 * A387214 A387215 A387216

Keyword

nonn,cons,changed

Author

Amiram Eldar, Aug 22 2025

Status

approved