# Indefinite Integration

In calculus, associate degree antiderivative, primitive perform, primitive integral or indefinite integral[Note 1] of a perform f could be a differentiable perform F whose by-product is adequate to the first perform f. this will be explicit symbolically as .[1][2] the method of determination for antiderivatives is named antidifferentiation (or indefinite integration) and its opposite operation is named differentiation, that is that the method of finding a by-product.

Antiderivatives square measure associated with integrals through the basic theorem of calculus: the definite integral of a perform over associate degree interval is adequate to the distinction between the values of associate degree antiderivative evaluated at the endpoints of the interval.The separate equivalent of the notion of antiderivative is antidifference.

Non-continuous functions will have antiderivatives. whereas there square measure still open queries during this space, it's renowned that:Some extremely pathological functions with massive sets of discontinuities could withal have antiderivatives.

In some cases, the antiderivatives of such pathological functions is also found by Georg Friedrich Bernhard Riemann integration, whereas in alternative cases these functions aren't Georg Friedrich Bernhard Riemann integrable.Assuming that the domains of the functions square measure open intervals:

A necessary, however not spare, condition for a perform f to own associate degree antiderivative is that f have the intermediate worth property. That is, if [a, b] could be a subinterval of the domain of f and y is any complex number between f(a) and f(b), then there exists a c between a and b specified f(c) = y. this is often a consequence of Darboux's theorem.The set of discontinuities of f should be a hardscrabble set. This set should even be associate degree F-sigma set (since the set of discontinuities of any perform should be of this type).

Moreover, for any hardscrabble F-sigma set, one will construct some perform f having associate degree antiderivative, that has the given set as its set of discontinuities.If f has associate degree antiderivative, is delimited on closed finite subintervals of the domain and contains a set of discontinuities of

Lebesgue live zero, then associate degree antiderivative is also found by integration within the sense of Lebesgue. In fact, exploitation a lot of powerful integrals just like the Henstock–Kurzweil integral, each perform that associate degree antiderivative exists is integrable, and its general integral coincides with its antiderivative.

If f has associate degree antiderivative F on a bounded interval [a,b], then for any alternative of partition <\dots =b} a=x_<\dots =b, if one chooses sample points ^\in [x_,x_]} x_^\in [x_,x_] as such by the norm theorem, then the corresponding Georg Friedrich Bernhard Riemann add telescopes to the worthFinding antiderivatives of elementary functions is commonly significantly tougher than finding their derivatives. for a few elementary functions, it's not possible to seek out associate degree antiderivative in terms of alternative elementary functions. See the articles on elementary functions and nonelementary integral for any info.

There square measure varied ways available:the dimensionality of integration permits United States to interrupt difficult integrals into less complicated ointegration by substitution, typically combined with pure mathematics identities or the Napierian logarithm.the inverse chain rule methodology, a special case of integration by substitution .Integration by components to integrate merchandise of functions

Inverse function integration, a formula that expresses the associate degreetiderivative of the inverse } f^ of an invertible and continuous perform f in terms of the antiderivative of f and of } f^.the method of partial fractions in integration permits United States to integrate all rational functions (fractions of 2 polynomials)

the Risch algorithmic programwhen group action multiple times, sure further techniques are often used, see as an example double integrals and polar coordinates, the Jacobian and also the Stokes' theoremif a perform has no elementary antiderivative (for instance, {\displaystyle \exp(-x^)} {\displaystyle \exp(-x^)}), its integral are often approximated exploitation numerical integrationt is typically convenient to algebraically manipulate the integrand specified alternative integration techniques, like integration by substitution, is also used.to calculate the (n times) perennial antiderivative of a perform f

Cauchy's formula is helpful (cf. Cauchy formula for perennial integration):{\displaystyle \int _{x_}^\int _{x_}^{x_}\dots \int _{x_}^}f(x_)\,dx_\dots \,dx_\,dx_=\int _{x_}^f(t)}}\,dt.} {\displaystyle \int _{x_}^\int _{x_}^{x_}\dots \int _{x_}^}f(x_)\,dx_\dots \,dx_\,dx_=\int _{x_}^f(t)}}\,dt.}

Computer pure mathematics systems are often wont to automatize some or all of the work conc