An integral more general than the SCP-integral is introduced, and among other properties, it is proved that iff is integrable in this sense with its second primitive ACG* and ifG is the indefinite integral of a functiong of bounded variation, thenfG is also integrable in this sense and the integration by parts formula holds. The integration by parts formula for the SCP-integral is also proved wheng is of bounded variation. This advances the existing result when, in addition,g is also continuous.