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Suppose I have a real-valued function in two variables $f$, and another function $$F(t) = \int_a^b f(x, t) \, \text{d}x$$ for some real numbers $a < b$.

A commonly used trick is differentiating under the integral sign, which states that for "nice functions" $f$, we have $$F'(t) = \int_a^b \frac{\partial f}{\partial t} \,\text{d}x$$

(in other words, we can interchange the differentiation and integration operators).

My question is: under what assumptions for $f$ does the above result hold? In those cases, how can we prove that differentiation under the integral sign works?

(I have heard that this type of result is discussed in Lang's Undergraduate Analysis, but unfortunately I do not have access to that book)

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  • $\begingroup$ This sounds like asking when we can do $$\lim \int f_n = \int f, \int \int dy dx = \int \int dx dy, \int \int d \mu d \lambda = \int \int d \lambda d \mu, \frac{\partial^2}{\partial y \partial x} = \frac{\partial^2}{\partial x \partial y}$$: These are very big and common questions in math, and there are a lot of answers. For instance, Integrating x^2 e^{-x} using Feynman's trick $\endgroup$ – user198044 Oct 11 '18 at 1:18
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    $\begingroup$ Usually this is done using variants of the dominated convergence theorem with the domination on the derivative. $\endgroup$ – Delta-u Oct 11 '18 at 16:38

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