

A265011


Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.


1



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
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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 TrigonometricLogarithmic Definite Integrals, in the Mathematica Journal, Vol. 19.10 (2017).
W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were twosided, half were onesided.) See page 8.


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.
Cf. A309209 (continued fraction of the negation of this constant).
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



