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We will identify sufficient conditions that allow us to guarantee different kinds of stabilities for a class of integral equations with kernels depending on sine and cosine functions. Conditions will be identified to ensure Hyers-Ulam stability, Hyers-Ulam-Rassias stability and other intermediate stabilities for those kind of integral equations. This will be done by taking profit of fixed point arguments in the framework of spaces of continuous functions endowed with generalizations of the Bielecki metric.