In the weak 16th Hilbert problem, the Poincar´e-Pontryagin-Melnikov function, M1(h); is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this talk we shall present a compact expression for the first-order Taylor series of the function M1(h; a) with respect to a, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on a: We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples. This is a joint work with Jackson Itikawa (UNIR, Brazil) and Joan Torregrosa (UAB, Spain).