A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.
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
Examples
This integral is equal to 0.50667090321662298198525580478358151247...
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.
Crossrefs
Programs
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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}] -
PARI
intnum(x=0,1,sin(log(x))/(x+1)/log(x))
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)).
Comments