(* 4.381.2 *)

Integrate[Log[x]*Cos[a*x],{x,0,1},Assumptions->Element[a,Reals]]

f1[a_]:=NIntegrate[Log[x]*Cos[a*x],{x,0,1}];

f2[a_]:=N[-((SinIntegral[a])/a)];

f1[1]

f2[1]

f[a_]:=f1[a]/f2[a];

f[RandomReal[{0,20},10]]

(* Correct *)

(* Be careful with the notation: GR has si and Si and they differ by Pi/2 *)