Given a projective non-singular curve of genus g and an integer \(k\geq 2g- 1\), the natural map from the k-symmetric product of the curve to the jacobian is a projective bundle. The pull-back of this bundle to the universal cover of the jacobian is trivial and it should be possible to describe the bundle by means of multiplicative factors. This is the problem of inversion of abelian integrals proposed by Kempf. In this paper, an explicit description of the bundle is given (in case \(k=2g-1)\) in terms of trivializing sets and transition functions. This generalizes a result of \textit{K. R. Coombes} and \textit{R. J. Fisher} [Math. Ann. 275, 185-196 (1986; Zbl 0579.14023)] which deals only with the cases \(g=2\) and \(g=3\) for non-hyperelliptic curves.