In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.
The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments , including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".
In mathematics, an elementary function is a function of a single variable that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation; it is indeed derived using the product rule.
In calculus, the constant of integration, often denoted by , is a constant term added to an antiderivative of a function to indicate that the indefinite integral of , on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.
In mathematics, an expression or equation is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.
In mathematics, differential Galois theory is the field that studies extensions of differential fields.
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that
In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. Roughly speaking, the two operations can be thought of as inverses of each other.
In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions.
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