Fundamental Theorem Of Calculus

Statement

The Fundamental Theorem of Calculus (FTC) links the concept of differentiating a function with the concept of integrating a function.

Part 1

If $f$ is continuous on $[a,b]$, then the function $F(x) = \int_{a}^{x} f(t)dt$ is differentiable on $(a,b)$ and $F'(x) = f(x)$.

Part 2

If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then: $\int_{a}^{b} f(x)dx = F(b) - F(a)$

Relationships

• Related: Calculus Derivatives, Integral
• Topic: Calculus