Antiderivative in the context of Differentiable function

In mathematics, an integral is the continuous analog of a sum, and 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.

A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be thought of as inverses of each other.

The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound.

In differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians, including Leibniz, Newton, Lagrange, and Arbogast. The usefulness of each notation depends on the context in which it is used, and it is sometimes advantageous to use more than one notation in a given context. For more specialized settings—such as partial derivatives in multivariable calculus, tensor analysis, or vector calculus—other notations, such as subscript notation or the ∇ operator are common. The most common notations for differentiation (and its opposite operation, antidifferentiation or indefinite integration) are listed below.

The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives.

In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as the absolute value function and piecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary.

This list of elementary functions was originally set forth by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function ${\displaystyle {\sqrt {z}}}$ or ${\displaystyle \log z}$) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.

In calculus, the constant of integration, often denoted by ${\displaystyle C}$ (or ${\displaystyle c}$), is a constant term added to an antiderivative of a function ${\displaystyle f(x)}$ to indicate that the indefinite integral of ${\displaystyle f(x)}$ (i.e., the set of all antiderivatives of ${\displaystyle f(x)}$), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.

More specifically, if a function ${\displaystyle f(x)}$ is defined on an interval, and ${\displaystyle F(x)}$ is an antiderivative of ${\displaystyle f(x),}$ then the set of all antiderivatives of ${\displaystyle f(x)}$ is given by the functions ${\displaystyle F(x)+C,}$ where ${\displaystyle C}$ is an arbitrary constant (meaning that any value of ${\displaystyle C}$ would make ${\displaystyle F(x)+C}$ a valid antiderivative). For that reason, the indefinite integral is often written as ${\textstyle \int f(x)\,dx=F(x)+C,}$ although the constant of integration might be sometimes omitted in lists of integrals for simplicity.