Antiderivative
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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:
- the linearity of integration allows us to break complicated integrals into simpler ones,
- integration by substitution, often combined with trigonometric identities or the natural logarithm,
- integration by parts to integrate products of functions,
- the inverse chain rule method, a special case of integration by substitution,
- the method of partial fractions in integration allows us to integrate all rational functions (fractions of two polynomials),
- the Risch algorithm,
- integrals can also be looked up in a table of integrals,
- when integrating multiple times, we can use certain additional techniques, see for instance double integrals and polar coordinates, the Jacobian and the Stokes theorem,
- computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy,
- if a function has no elementary antiderivative (for instance, exp(x²)), its integral can be approximated using numerical integration.
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