@MISC{INTEGR26.MIZ,
  AUTHOR = {Endou, Noboru},
  TITLE = {Antiderivatives and Integration},
  DAY = {30},
  MONTH = {June},
  YEAR = {2023},
  ADDRESS1 = {National Institute of Technology, Gifu College\\ 2236-2 Kamimakuwa, Motosu, Gifu, Japan},
  SECTION1 = {Basic Theorems on Continuity and Derivativity},
  SECTION2 = {Generalization of Previous Theorems},
  SECTION3 = {Antiderivatives and Related Theorems},
  SUMMARY = {In this paper, we introduce indefinite integrals (antiderivatives) and proof
  integration by substitution.
  In our previous article \cite{INTEGRA7.abs}, we have introduced an indefinite-like integral,
  but it is incorrect because it must be an integral over the whole set of real numbers.
  For this reason, to define the antiderivative for a function, we use the derivative of an
  arbitrary interval \cite{FDIFF_12.abs}. Furthermore, antiderivatives are also used to modify
  the integration by substitution and integration by parts. In the first section, we summarize
  the basic theorems on continuity and derivativity. In the second section, we generalize some
  theorems that were noticed during the formalization process. In the last section, we define
  the antiderivatives and formalize the integration by substitution and the integration by parts.
  We referred to \cite{Apostol:1967} and \cite{Courant:1988} in the formalization.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 23K11242.}}

@BOOK{Apostol:1967,
 AUTHOR = {Apostol, Tom M.},
 TITLE = {Calculus},
 PUBLISHER = {John Wiley \& Sons},
 YEAR = {1967},
 VOLUME = {I},
 EDITION = {Second}}

@BOOK{Courant:1988,
 AUTHOR = {R. Courant, E.J. McShane},
 TITLE = {Differential and Integral Calculus},
 PUBLISHER = {John Wiley \& Sons},
 YEAR = {1988}}

