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 A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx. 3
 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 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: 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 September 16 23:53 EDT 2021. Contains 347477 sequences. (Running on oeis4.)