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Fundamental Theorem of Calculus

Antiderivative, inverse derivative, primitive function or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.



For lists of antiderivatives of primitive functions, see lists of integrals.

Definite integral f(x) = sin x from 0 to x

Drag the blue point x to change the upper limit of the definite integral. Observe how the area changes with x.

Definite integral f(x) = x from 0 to x

Integral with singularity

Integral with boundary singularity

Integrfal from 0 to ∞ could be finite un infinite value. Using the Newton-Leibniz formula and finding the limiting value for upper boundary give us: