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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a principle that links the concept of the derivative of a function with the concept of an integral. It serves as a central tenet in calculus, revealing the relationship between differentiation and integration. Mathematically, it states that if a function is continuous over a closed interval, the integrated result over this interval can be expressed in terms of antiderivatives.

For example, if \(f\) is continuous on \([a, b]\) and \(F\) is an antiderivative of \(f\) on \([a, b]\), then:

\(\int_{a}^{b} f(x)\,dx = F(b) - F(a)\)