Derivation of recurrence formulas for general many-electron overlap, nuclear attraction, and electron repulsion integrals over explicitly correlated Cartesian Gaussian functions is presented. The recurrence formulas are derived in a similar way as the derivation of molecular two-electron integrals over Cartesian Gaussian functions by Obara and Saika. As a result, the formulas expressing the many-electron integrals of higher angular momentum as a linear combination of those of lower angular momentum are obtained. An algorithm for computation of the general many-electron integrals by means of the recurrence formulas is also shown.