The article provides detailed information about the concept of integral and the history of its origin. In addition to the Riemann integral, which is covered in the school program, a number of other integrals (Riemann-Stiltes, Daniell, Alfred Haar, Henstock-Kurzweil, Ito and Stratonovich, Chokvet) are described. The difference between integrals is explained with the help of concrete examples. The concept of definite integral and its application in practical problems are widely covered, counterexamples of some theorems are presented