# Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a central result in calculus that establishes a connection between differentiation and integration. It consists of two parts that demonstrate the inverse relationship between these two essential operations.

The first part of the FTC states that if a function is continuous on a closed interval [`a, b`

], and its antiderivative (also called the indefinite integral) is `F(x)`

, then the definite integral of the function over the interval [`a, b`

] is equal to the difference in the values of the antiderivative at the endpoints of the interval: `∫[a, b] f(x) dx = F(b) - F(a)`

.

The second part of the FTC states that if a function is continuous on an interval [`a, b`

] and has an antiderivative `F(x)`

on that interval, then the derivative of the antiderivative with respect to `x`

is equal to the original function: `d/dx [∫[a, x] f(t) dt] = f(x)`

.

The Fundamental Theorem of Calculus has profound implications in mathematics, physics, and engineering. It allows us to find the area under a curve, solve problems involving motion and change, and provides a foundation for many advanced topics in calculus, such as Taylor series, integrals, and the Mean Value Theorem.

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