(* 4.384.2 *)

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

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

FullSimplify[-((EulerGamma+2 HarmonicNumber[-3-2 n]-HarmonicNumber[-2-n]+HarmonicNumber[-(1/2)-n]+HarmonicNumber[(1/2)+n]-PolyGamma[0,n]+2 PolyGamma[0,2 n])/(2 π+4 n π))]

(* How does one simplify this? *)

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

f2[n_]:=N[2*(Log[2]-1/(2*n+1)-2*Sum[1/(2*k-1),{k,1,n}])/((2*n+1)*Pi)];

Table[f1[n]/f2[n],{n,1,6}]

(* Correct !! *)