(*4.381.1*)

Integrate[Log[x]*Sin[a*x],{x,0,1},Assumptions->a>0]

(* This matches the answer in the table *)

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

f2[a_]:=N[-((EulerGamma-CosIntegral[a]+Log[a])/a)];

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

f[RandomReal[{0,100},30]]

(* Correct *)