Antiderivative

From Academic Kids

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 finding antiderivatives is antidifferentiation (or indefinite integration).

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:

<math>\int_a^b f(x)\, dx = F(b) - F(a).<math>

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:

<math>\int f(x)\, dx.<math>

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:

<math>F(x) = \int_a^x f(t)\,dt.<math>

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 polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations). Examples of these are

<math>\int e^{x^2}\,dx,\qquad \int \frac{\sin(x)}{x}\,dx,\qquad \int\frac{1}{\ln x}\,dx.<math>

For more on these facts, see differential Galois theory.

Techniques of integration

Finding antiderivatives is considerably harder than finding derivatives. We have various methods at our disposal:

de:Stammfunktion es:Integral y funcin primitiva fr:Primitive (mathmatiques) ja:不定積分 ru:Первообразная lt:Pirmyktė funkcija

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