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
KEYWORD
cons,nonn
AUTHOR
John M. Campbell, Apr 06 2016
STATUS
approved