Fundamental Theorem of Calculus/First Part/Corollary

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Corollary to Fundamental Theorem of Calculus (First Part)

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $F$ be a real function which is defined on $\left[{a \,.\,.\, b}\right]$ by:

$\displaystyle F \left({x}\right) = \int_a^x f \left({t}\right) \rd t$


Then:

$\displaystyle \frac \d {\d x} \int_a^x f \left({t}\right) \rd t = f \left({x}\right)$


Proof

Follows from the Fundamental Theorem of Calculus (First Part) and the definition of primitive.

$\blacksquare$


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