The Fundamental Theorem of Calculus

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The Fundamental Theorem of Calculus

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The Definite Integral as Antiderivative

The other part of the Fundamental Theorem (sometimes called the first Fundamental Theorem) relates indefinite integrals to anti-derivatives:

(First) Fundamental Theorem of Calculus:

If $f$ is a continuous function, then $\frac{d}{dx} \int_a^x f(s)\, ds = f(x).$

That is, the indefinite integral is an anti-derivative. The derivative of the (indefinite) integral is the original function.

Warning: The notation $\int f(x) \,dx$ , without any upper and lower limits on the integral sign, is used in two different ways. Sometimes it is used to mean the indefinite integral $\int_a^x f(s) \,ds$ , and sometimes it is used to mean "an anti-derivative of $f(x)$ ". Since $\int_a^x f(s)\, ds$ is an anti-derivative, this is consistent.