AP Calculus AB and BC glossary

Fundamental Theorem of Calculus Part 1

Also called: FTC Part 1

Part 1 of the Fundamental Theorem of Calculus says that differentiating an integral with a variable upper bound returns the integrand evaluated at that bound. It is the statement that differentiation and integration are inverse operations.

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

The lower bound being constant is what makes this work; it contributes nothing to the derivative. If the upper bound is a function of $x$ , the chain rule adds a factor of its derivative.

The mistake

Forgetting the chain rule factor. The derivative of $\int_0^{x^2} f(t)\,dt$ is $f(x^2) \cdot 2x$ , not $f(x^2)$ .

Appears in: Unit 6: Integration and Accumulation