In this paper, we shall establish some fixed point theorems for mappings with the contractive condition of integrable type on complete intuitionistic fuzzy metric spaces (X;M;N; *; ♢). We also use Lebesgue-integrable mapping to obtain new results. Akram, Zafar, and Siddiqui introduced the notion of A-contraction mapping on metric space. In this paper by using the main idea of the work, we introduce the concept of A-fuzzy contractive mappings. Finally, we support our results by some examples.