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For example: F(x) = x³ / 3 is an antiderivative of f(x) = x². As the derivative of a constant is zero, x² will have an infinite number of antiderivatives; such as (x³ / 3) + 0 and (x³ / 3) + 7 and (x³ / 3) - 36...thus; the antiderivative family of x² is collectively referred to by F(x) = (x³ / 3) + C; where C is any constant. Essentially, related antiderivatives are vertical translations of each other; each graph's location depending upon the value of C.

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 integrationIn calculus, the indefinite integral of a given function (i. the set of all antiderivatives of the function) is always written with a constant, the constant of integration . This constant expresses an ambiguity inherent in the construction of antiderivati.

Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary:

This is another formulation of the fundamental theorem of calculus.

There are also some non-continuous 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 polynomialIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives os, exponential functionThe exponential function is one of the most important functions in mathematics. It is written as exp x or e x where e is the base of the natural logarithm. As a function of the real variable x the graph of e x is always positive (above the x axis) and incs, logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are rs, trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios ofs, inverse trigonometric functions and their combinations). Examples of these are

For more on these facts, see differential Galois theory