(* 4.384.1 *)

Integrate[Log[Sin[Pi*x]]*Sin[2*n*Pi*x],{x,0,1}]

Integrate[Log[Sin[Pi*x]]*Sin[2*n*Pi*x],{x,0,1},Assumptions->Element[n,Integers]]

(* This is incorrect !! *)

f1[n_]:=NIntegrate[Log[Sin[Pi*x]]*Sin[2*n*Pi*x],{x,0,1}];

Table[f1[n],{n,1,6}]

(* Correct !! *)