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.


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