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After a short historical overview of integration and recalling basic integrals of Riemann and Lebesque, we will look on representation of special functionals acting on measurable non-negative functions, yielding special types of integrals, such as Choquet or Sugeno integrals. Based on particular decompositions of considered functions, and on a fixed monotone measure, the class of decompositions integrals will be discussed. A general framework for integrals with respect to monotone measures, universal integrals, will be also clarified and exemplified. A special stress will be put on integrals based on copulas.