# Fundamental theorem of calculus

The **fundamental theorem of calculus** is a theorem that links the concept of differentiating a function (calculating the gradient) with the concept of integrating a function (calculating the area under the curve). The two operations are inverses of each other apart from a constant value which depends where one starts to compute area.

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The first part of the theorem, sometimes called the **first fundamental theorem of calculus**, states that one of the antiderivatives (also known as an *indefinite integral*), say *F*, of some function *f* may be obtained as the integral of *f* with a variable bound of integration. This implies the existence of antiderivatives for continuous functions.[1]

Conversely, the second part of the theorem, sometimes called the **second fundamental theorem of calculus**, states that the integral of a function *f* over some interval can be computed by using any one, say *F*, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals.