Newton-Leibniz formula

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The formula expressing the value of a definite integral of a given function $f$ over an interval as the difference of the values at the end points of the interval of any primitive (cf. Integral calculus)$F$ of the function $f$: \begin{equation}\label{eq:*} \int\limits_a^bf(x)\,dx = F(b)-F(a). \end{equation} It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later.

If $f$ is Lebesgue integrable over $[a,b]$ and $F$ is defined by \begin{equation*} F(x) = \int\limits_a^xf(t)\,dt + C, \end{equation*} where $C$ is a constant, then $F$ is absolutely continuous, $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid.

A generalization of the Newton–Leibniz formula is the Stokes formula for orientable manifolds with a boundary.


The theorem expressed by the Newton–Leibniz formula is called the fundamental theorem of calculus, cf. e.g. [a1].


[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff
[a2] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 165ff
How to Cite This Entry:
Newton-Leibniz formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article