Differentiation of integral

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Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):

Calculate:
[tex]
\frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt
[/tex]
 

Answers and Replies

  • #2
HallsofIvy
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Dear all, a question which has puzzled me for some days:
(Assume that all are differentiable enough times):

Calculate:
[tex]
\frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt
[/tex]
Leibniz's rule, which generalizes the fundamental theorem of Calculus:
[tex]\frac{d}{dx} \int_{g(x)}^{h(x)} f(x,t)dt= \frac{dh}{dx}f(x, h(x))- \frac{dg}{dx}f(x,g(x))+ \int_{g(x)}^{h(x)} \frac{\partial f(x,t)}{\partial x} dt[/tex]
 

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