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A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx. 0
5, 0, 6, 6, 7, 0, 9, 0, 3, 2, 1, 6, 6, 2, 2, 9, 8, 1, 9, 8, 5, 2, 5, 5, 8, 0, 4, 7, 8, 3, 5, 8, 1, 5, 1, 2, 4, 7, 2, 8, 4, 3, 5, 4, 7, 3, 4, 7, 0, 2, 0, 5, 8, 2, 9, 2, 0, 0, 0, 2, 4, 5, 8, 6, 5, 9, 4, 7, 0, 5, 1, 4, 5, 1, 3, 2, 2, 6, 9, 3, 1, 5, 0, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This integral has an elegant evaluation in terms of the gamma function (see below formula). There is an interesting "symmetry" between the expressions involving the gamma function in this evaluation.

LINKS

Table of n, a(n) for n=0..83.

John M. Campbell, An Algorithm for Trigonometric-Logarithmic Definite Integrals, in the Mathematica Journal, Vol. 19.10 (2017).

FORMULA

Equals log(2) + log(((Gamma(1 - i/2)^2*Gamma(1 + i))/(Gamma(1 + i/2)^2*Gamma(1 - i)))^(i/2)), where i = sqrt(-1) denotes the imaginary unit.

Equals Sum_{n >= 0} (-1)^n*arctan(1/(n+1)).

EXAMPLE

This integral is equal to 0.50667090321662298198525580478358151247...

MATHEMATICA

Print[RealDigits[Re[Log[2] + Log[((Gamma[1 - I/2]^2 Gamma[1 + I])/(Gamma[1 + I/2]^2 Gamma[1 - I]))^(I/2)]], 10, 100]] ;

NIntegrate[Sin[Log[x]]/(x + 1)/Log[x], {x, 0, 1}]

PROG

(PARI) intnum(x=0, 1, sin(log(x))/(x+1)/log(x))

CROSSREFS

Decimal expansions of definite integrals over elementary functions: A256127, A256128, A256129, A204067, A204068, A205885, A206161, A206160, A206769, A229174, A083648, A094691, A098687, A177218, A188141, A233382, A256273, A258086,

Sequence in context: A144483 A098403 A166126 * A320375 A200419 A271522

Adjacent sequences:  A265008 A265009 A265010 * A265012 A265013 A265014

KEYWORD

cons,nonn

AUTHOR

John M. Campbell, Apr 06 2016

STATUS

approved

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Last modified December 18 20:06 EST 2018. Contains 318245 sequences. (Running on oeis4.)