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A265011
Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.
4
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
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
John M. Campbell, An Algorithm for Trigonometric-Logarithmic 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 two-sided, half were one-sided.) 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: A098403 A353876 A166126 * A320375 A361918 A200419
KEYWORD
cons,nonn
AUTHOR
John M. Campbell, Apr 06 2016
STATUS
approved