LEIBNIZ-NEWTON FORMULA AND ITS GENERALIZATIONS

V diplomskem delu obravnavamo Leibniz-Newtonovo formulo in njene posplošitve. Pojem Riemannovega oz. določenega integrala je vpeljan s pomočjo Darbouxovih in Riemannovih vsot, pri čemer je poudarjena ekvivalentnost omenjenih pristopov. V tretjem poglavju so predstavljeni potrebni pogoji za integrabilnost funkcij, v četrtem pa ena izmed povezav med določenimi integrali in primitivnimi funkcijami. Sledi pomemben matematični rezultat Leibniza in Newtona, t. i. Leibniz-Newtonova formula, poznana tudi kot osnovni izrek analize. V nadaljevanju obravnavamo posplošitve te formulepri prvi obliki nadomestimo obojestranski odvod z desnim odvodom, v drugi nastopa Schwarzov odvod, tretja posplošitev Leibniz-Newtonove formule pa se nanaša na funkcije, ki izpolnjujejo Lipschitzev pogoj. V zadnjem poglavju je navedenih nekaj primerov, kjer postane računanje določenega integrala s pomočjo Leibniz-Newtonove formule precej lažje kot računanje po definiciji določenega integrala. Ključna sta primera, ki ponazarjata napake, ki nastanejo pri uporabi rešitev iz tablic nedoločenih integralov na neustreznih intervalih. Pokazali smo, kako se tem napakam uspešno izogniti.The graduation thesis discusses the Leibniz-Newton formula and its generalizations. The notion of Riemann or the definite integral is introduced by presenting Darboux and Riemann sums, whereby the equivalence of these approaches is emphasised. The third chapter presents the necessary conditions for integrability of functions and in the fourth chapter a connection of the definite integral and the primitive function is presented. This is followed by an important mathematical result, the Leibniz-Newton formula, also known as the fundamental theorem of calculus. Several generalizations of this formula are then discussedin the first form we assumed that f is only the right-hand derivative of F, the second form uses the Schwarz derivative, and the third generalization of the Leibniz-Newton formula refers to the functions that satisfy the Lipschitz condition. In the last chapter some examples for calculating the definite integrals using the Leibniz-Newton formula are given, and it is much easier to do so, than to calculate them by using the definition of the definite integral. The most important among them are two examples that illustrate mistakes that occur when using solutions from the table of indefinite integrals at unsuitable intervals. We have shown how to successfully avoid making such mistakes