Antiderivative

In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i.e., F′ = f. The process of solving for antiderivatives is antidifferentiation (or indefinite integration).

Related Topics:
Calculus - Real valued - Function - Derivative

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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.

Related Topics:
Constant - Zero - Infinite - Family - Vertical translation - Graph's - Value

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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:

Related Topics:
Integrals - Fundamental theorem of calculus

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:int_a^b f(x), dx = F(b) - F(a).

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Because of this, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f and is written using the integral symbol with no bounds:

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:int f(x), dx.

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But it is critical to remember that an integral isn't the same, in general, as the means for evaluating it; and the function that an integral implies stands apart from that means - in the case of single-variable integrals, from antiderivatives.

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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.

Related Topics:
Interval - Arbitrary constant of integration

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

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:F(x) = int_a^x f(t),dt.

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Varying the lower boundary produces other antiderivatives. This is another formulation of the fundamental theorem of calculus.

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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.

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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

Related Topics:
Elementary function - Polynomial - Exponential function - Logarithm - Trigonometric function - Inverse trigonometric function

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:int e^{x^2},dx,qquad int rac{sin(x)}{x},dx,qquad intrac{1}{ln x},dx.

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For more on these facts, see differential Galois theory.

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