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Fundamental Theorem of Calculus

There are two parts to it

Suppose that the function ff is continuous on an interval II containing the point aa.

Part 1

Let the function FF be defined on II:

F(x)=axF(t)dt=axf(t)dtF(x) = \int_{a}^{x} F'(t) dt = \int_{a}^{x} f(t) dt

Then F is differentiable on II, and F(x)=f(x)F'(x) = f(x). Thus FF is an antiderivative of ff on the interval II.

Part 2

If G(x)G(x) is any antiderivative of f(x)f(x) on II such that G(x)=f(x)G'(x) = f(x) on II, then for any bb in II we have:

abf(x)dx=G(b)G(a)\int_{a}^{b} f(x) dx = G(b) - G(a)

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