 # Fundamental Theorem of Calculus

First fundamental theorem of integral calculus

Let the area function be defined by A (x) = \int_{a}^{x} f(x) dx for all x ≥ a, where the function f is assumed to be continuous on [a, b]. Then A' (x) = ƒ(x) for all x ∈ [a, b].

Second fundamental theorem of integral calculus

Let ƒ be a continuous function of x defined on the closed interval [a, b] and let F be another function such that \frac{d}{dx} F(x) = f(x) for all x in the domain of ƒ, then \int_{a}^{b} f(x) dx = \left[F(x) + C \right]_{a}^{b} = F(b) - F(a).

This is called the definite integral of ƒ over the range [a, b], where a and b are called the limits of integration, a being the lower limit and b the upper limit.

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•  \int_{a}^{a} f(x) dx = 0
• \tt \int f(x) dx = F(x) + c \ then \ \ \int_{a}^{b} f(x) dx = F(b) - F(a)