I have proved
Theorem. Let f(n)=integrate(sin(x)/x^(2*n),x,1,inf). Then
f(1)=sin(1) - expintegral_ci(1),
f(n) = -f(n-1)/(2*(n-1)*(2*n-1))+cos(1)/(2*(n-1)*(2*n-1))+sin(1)/(2*n-1)
(%i1) f(n):=if n=1 then sin(1)-expintegral_ci(1) else
-f(n-1)/(2*(n-1)*(2*n-1))+cos(1)/(2*(n-1)*(2*n-1))+sin(1)/(2*n-1)$
Next we compute integrate(sin(x)/x^(2*n),x,1,inf) for n=2, 3, 4, 5, ....
(%i2) f(2),ratsimp;
(%o2) (sin(1)+expintegral_ci(1)+cos(1))/6
(%i3) float(%), numer;
(%o3) 0.28652953559617
(%i4) f(3),ratsimp;
(%o4) (23*sin(1)-expintegral_ci(1)+5*cos(1))/120
(%i5) float(%), numer;
(%o5) 0.18098283547518
(%i6) f(4),ratsimp;
(%o6) (697*sin(1)+expintegral_ci(1)+115*cos(1))/5040
(%i7) float(%), numer;
(%o7) 0.12876536617239
(%i8) f(5),ratsimp;
(%o8) (39623*sin(1)-expintegral_ci(1)+4925*cos(1))/362880
(%i9) float(%), numer;
(%o9) 0.099212566918874
This it's the same as numeric results from
http://www.math.utexas.edu/pipermail/maxima/2012/027977.html
I thank Edwin Woollett for the interesting problem.
Aleksas D.