A389356 Decimal expansion of Integral_{x=0..Pi/2} {sqrt(tan(x))} dx, where {} denotes fractional part.

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

Offset

0,1

Comments

The integral without the sqrt function is 1 - A212880.

The integral without the fraction part function is A247719.

Links

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

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 1.35, "A Surprisingly Awesome Fractional Part Integral with Forms Involving sqrt(tan(x)) and sqrt(cot(x))", p. 48, section 2.34, p. 83, and section 3.35, pp. 270-274.

Formula

Equals Integral_{x=0..Pi/2} {sqrt(cot(x))} dx, where {} denotes fractional part.

Equals (2*sqrt(2)-1) * Pi/4 + arctan(tanh(Pi/sqrt(2))/tan(Pi/sqrt(2))).

Example

0.79669969064920223374639268814111269650524451008602...

Mathematica

RealDigits[(2*Sqrt[2]-1) * Pi/4 + ArcTan[Tanh[Pi/Sqrt[2]]/Tan[Pi/Sqrt[2]]], 10, 120][[1]]

Prog

(PARI) (2*sqrt(2)-1) * Pi/4 + atan(tanh(Pi/sqrt(2))/tan(Pi/sqrt(2)))

Crossrefs

Cf. A212880, A247719, A352527.

Sequence in context: A200098 A239069 A154168 * A019644 A388457 A100639

Adjacent sequences: A389353 A389354 A389355 * A389357 A389358 A389359

Keyword

nonn,cons

Author

Amiram Eldar, Oct 01 2025

Status

approved