(* 4.381.3 *)

Integrate[Log[x]*Sin[n*x],{x,0,2*Pi},Assumptions->Element[n,Integers]&&n>0]//FullSimplify

(* How can we simplify this? *)

f1[n_]:=NIntegrate[Log[x]*Sin[n*x],{x,0,2*Pi}];

f2[n_]:=N[-((2 EulerGamma-2 CosIntegral[2 n π]+Log[4 (n^2) (π^2)])/(2 n))];

f[n_]:=f1[n]/f2[n];

f[RandomInteger[{0,200},10]]

(* Correct *)