Deriving Generalized Leibniz’s Integral Rule

The general form of Leibniz’s Integral Rule with variable limits states:

Suppose f(x, t) satisfies the condition stated previously for the basic form of Leibniz’s Rule (LR-1, see “A Semi-Rigorous Derivation of Leibniz’s Rule“) . In addition, a(t), b(t) are defined and have continuous derivatives for t_1\le t\le t_2. Then for t_1\le t \le t_2,

\frac{d}{d t}\int\limits_{a(t)}^{b(t)}f(x, t)\;dx = f(b(t),t)\cdot b'(t) -f(a(t),t)\cdot a'(t)+ \int\limits_{a(t)}^{b(t)}\frac{\partial}{\partial t}f(x, t)\;dx.\quad\quad\quad(1)

(1) can be derived as a consequence of LR-1, the Multivariable Chain Rule, and the Fundamental Theorem of Calculus (FTC):

Clearly,

\int\limits_{a(t)}^{b(t)}f(x, t)\;dx\quad\quad\quad(2)

on the left side of (1) is a function of t.

Let

u = a(t),\quad\quad\quad(3)

v = b(t),\quad\quad\quad(4)

w = t,\quad\quad\quad(5)

(2) can be expressed as

G(u,v,w) = \int\limits_{u}^{v}f(x,w)\;dx.

Hence, by the chain rule,

\frac{d}{d t}\int\limits_{u}^{v}f(x, w)\;dx = \frac{\partial}{\partial u}G(u,v,w)\cdot \frac{du}{dt} +  \frac{\partial}{\partial v} G(u,v,w)\cdot \frac{dv}{dt} +  \frac{\partial}{\partial w}G(u,v,w)\cdot \frac{dw}{dt}

where

\frac{\partial}{\partial u}G(u,v,w)=\frac{\partial}{\partial u}\int\limits_{u}^{v}f(x, w)\;dx

= \frac{\partial}{\partial u}\left(-\int\limits_{v}^{u}f(x, w)\;dx\right)

= -\frac{\partial}{\partial u}\int\limits_{v}^{u}f(x, w) \;dx

\overset{\textbf{FTC}}{=} -f(u, w)

\overset{(3), (5)}{=} -f(a(t), t),

\frac{\partial}{\partial v}G(u,v,w)=\frac{\partial}{\partial v}\int\limits_{u}^{v}f(x, w)\;dx \overset{\textbf{FTC}}{=}f(v, w) \overset{(4), (5)}{=} f(b(t), t)

and,

\frac{\partial}{\partial w}G(u,v,w)=\frac{\partial}{\partial w}\int\limits_{u}^{v}f(x, w)\;dx \overset{\textbf{LR-1}}{=} \int\limits_{u}^{v}\frac{\partial}{\partial w}f(x, w)\;dx\overset{(3), (4), (5)}{=} \int\limits_{a(t)}^{b(t)}\frac{\partial}{\partial t}f(x, t)\;dx.

It follows that

\frac{d}{d t}\int\limits_{a(t)}^{b(t)}f(x, t)\;dx =-f(a(t),t)\cdot a'(t) + f(b(t),t)\cdot b'(t) + \int\limits_{a(t)}^{b(t)}\frac{\partial}{\partial t}f(x, t)\;dx,

i.e.,

\frac{d}{d t}\int\limits_{a(t)}^{b(t)}f(x, t)\;dx = f(b(t),t)\cdot b'(t) -f(a(t),t)\cdot a'(t)+ \int\limits_{a(t)}^{b(t)}\frac{\partial}{\partial t}f(x, t)\;dx.

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