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Antiderivatives are important because they can be used to compute integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:
Because of this connection, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f and is written as an integral without boundaries:
If F is an antiderivative of f and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x) = F(x) + C for all x. C is called the arbitrary constant of integration.
Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary:
There are also some noncontinuous functions which have an antiderivative, for example f(x) = 2x sin (1/x)  cos(1/x) with f(0) = 0 is not continuous at x = 0 but has the antiderivative F(x) = x² sin(1/x) with F(0) = 0.
There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions[?] and their combinations). Examples of these are
Finding antiderivatives is considerably harder than finding derivatives. We have various methods at our disposal:
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