The discussion centers on whether the equality of two integrals involving a function \( f(x) \) implies the equality of the integrals of two other functions \( g(x) \) and \( h(x) \). It is established that while the integrals \( \int_a^b f(x)g(x) \, dx \) and \( \int_a^b f(x)h(x) \, dx \) are equal, this does not necessarily mean that \( \int_a^b g(x) \, dx = \int_a^b h(x) \, dx \). The reasoning provided indicates that the integral of the difference \( F(x) = g(x) - h(x) \) being zero does not imply \( F(x) \equiv 0 \) without additional conditions on \( F(x) \). The thread concludes with a note that the original poster has initiated a new thread for further discussion on the same topic.