Leibniz's Integral Rule

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Theorem

Let $\map f {x, t}$, $\map a t$, $\map b t$ be continuously differentiable real functions on some region $R$ of the $\tuple {x, t}$ plane.


Then for all $\tuple {x, t} \in R$:

$\displaystyle \frac \rd {\rd t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \map f {t, \map b t} \frac {\rd b} {\rd t} - \map f {t, \map a t} \frac {\rd a} {\rd t} + \int_{\map a t}^{\map b t} \frac {\partial} {\partial t} \map f {x, t} \rd x$


Proof


Also known as

This is also referred to in some sources as Leibniz's Rule, but as this name is also used for a different result, it is necessary to distinguish between the two.


Also see

$\displaystyle \frac {\rd} {\rd t} \int_a^b \map f {x, t} \rd x = \int_a^b \frac {\partial} {\partial t} \map f {x, t} \rd x$


Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.


Sources