The evaluation process of the two integrals ∫ ࢘ ࢘ ∫ and ࢘ࢊ(࢘ࣅ−)࢞ࢋ ࢘ ࢙ࢉ Is ࢘ࢊ(࢘ࣅ−)࢞ࢋ ࢙࢘ recapitulated , using elementary knowledge from complex variables subject. This is done in a general manner, thence some special cases are worked out in full. Once ,this is done we turn to the application of these important integrals in some physical problems. In this concern, positive energy bound states is a vital example. Keywords: Integral, Physics, Important , Bound State, Complex Variables.