Fundamental Theorem of Calculus/First Part/Corollary

Corollary to Fundamental Theorem of Calculus (First Part)

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Let $F$ be a real function which is defined on $\closedint a b$ by:

$\ds \map F x = \int_a^x \map f t \rd t$

Then:

$\ds \frac \d {\d x} \int_a^x \map f t \rd t = \map f x$

Proof

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

$\blacksquare$

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4.4$