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Mathematics for Physics

205 Indefinite Integral

A function that undoes differentiation of another function that is used to solve equations involving derivatives.

definition [d] (Indefinite Integral = Antiderivative = Primitive) The most general antiderivative of \(f\), written

  • \(\displaystyle \int f(x)\, dx = F(x) + C\) ,

where \(\dfrac{dF}{dx} = f(x)\) and \(C\) is an arbitrary constant of integration.

where

  • \(f\) is the integrand.
  • \(F\) is any antiderivative of \(f\).
  • \(\dfrac{dF}{dx}\) is the derivative of \(F\) with respect to \(x\).
  • \(C\) is the constant of integration.
  • \(x\) is the independent variable.

Note:

  • the indefinite integral denotes a family of functions, not a single number.

205.1 References

  1. Stewart, J. Calculus. — indefinite integral as the most general antiderivative \(F(x)+C\).
  2. Aleksandrov, A. D., Kolmogorov, A. N., & Lavrent’ev, M. A. Mathematics: Its Content, Methods and Meaning. Dover, 1999. — primitive.
  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — primitive and antiderivative.