# Ngô Quốc Anh

## September 14, 2018

### Leibniz rule for proper integral with parameter whose limits also depends on the parameter

Filed under: Uncategorized — Ngô Quốc Anh @ 21:24

The following Leibniz integral rule is well-known

Theorem. Let $f(x, t)$ be a function such that both $f(x, t)$ and its partial derivative $f_x(x, t)$ are continuous in $t$ and $x$ in some region of the $(x, t)$-plane, including $a(x) \leqslant t \leqslant b(x)$, $x_0 \leqslant x \leqslant x_1$. Also suppose that the functions $a(x)$ and $b(x)$ are both continuous and both have continuous derivatives for $x_0 \leqslant x \leqslant x_1$. Then, for $x_0 \leqslant x \leqslant x_1$,

$\displaystyle \frac {d}{dx}\left(\int _{a(x)}^{b(x)}f(x,t)\,dt\right)=f\big (x,b(x)\big )b'(x)-f\big (x,a(x)\big) a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial f }{\partial x}}(x,t)\,dt.$

The purpose of this note is to show that, in fact, it is not  is not necessary to assume the function $f$ to be continuous. We note that this is indeed the case in which the limits of the integral $\int_{a(x)}^{b(x)}f(x,t)\,dt$ do not depend on the parameter $x$. For convenience, it is routine to assume the continuity, which immediately implies that all integrals are well-defined.

As mentioned above, we want to show that this is also the case for integrals of the form above.