A minimal integral, the \(C\)-integral, that will integrate all derivatives and all Lebesgue integrable functions, was defined by \textit{A. M. Bruckner, R. J. Fleissner} and \textit{J. Foran} [Colloq. Math. 50, 289-293 (1986; Zbl 0604.26006)], and later an ingenious constructive definition based on the Henstock Kurzweil approach was given by \textit{B. Bongiorno} [Matematiche 51, No. 2, 299-313 (1996; Zbl 0929.26007)], and \textit{B. Bongiorno, L. Di Piazza} and \textit{D. Preiss} [J. Lond. Math. Soc., II. Ser. 62, No. 1, 117-126 (2000; Zbl 0980.26006)]. The present paper continues the investigation of the properties of this integral: (i) a variational measure is defined in terms of which the indefinite \(C\)-integral is characterized; (ii) it is shown that all functions that are equivalent to functions of bounded variation, so-called BV functions, are multipliers for the \(C\)-integral; (iii) that the product of a derivative and a BV function is also a derivative up to a Lebesgue integrable function of arbitrarily small norm.