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Let $f$ be a function which satisfies the conditions of Fundamental theorem of calculus etc etc etc and $F(x) = \int_{a}^x f( \tau ) d \tau $. We know

$$ F'(x) = f(x) $$

Also, by the chain rule if we have

$$ F(x) = \int\limits_a^{g(x)} f ( \tau) d \tau $$

then $$ F'(x) = g'(x) f ( g(x) ) $$

My question: What about if

$$ F(x) = \int\limits_{h(x)}^{g(x)} f (t) dt $$

what would $F'(x)$ be? thanks

ILoveMath
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    Separate the last integral in the sum of two integrals, each of these with $a$ being one of the endpoints. – Git Gud Oct 14 '14 at 14:08

1 Answers1

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Hint

As said by @GitGud in comment write

$$F(x) = \int\limits_{h(x)}^{g(x)} f (t) dt= \int\limits_{h(x)}^{a} f (t) dt + \int\limits_{a}^{g(x)} f (t) dt=\int\limits_{a}^{g(x)} f (t) dt-\int\limits_{a}^{h(x)} f (t) dt.$$ Now, you can get the derivative easily.

mfl
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