In calculus, an antiderivative, primitive integral or indefinite integral

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry.

In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus.

John Wallis exploited an infinitesimal he denoted in area calculations, preparing the ground for integral calculus.

Infinitesimal calculus consists of differential calculus and integral calculus, respectively used for the techniques of differentiation and integration.

Several mathematicians, including Maclaurin and d ' Alembert, attempted to prove the soundness of using limits, but it would be 150 years later, through the work of Augustin Louis Cauchy and Karl Weierstrass, where a means was finally found to avoid mere " notions " of infinitely small quantities, that the foundations of differential and integral calculus were made firm.

Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change while integral calculus is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines and, such that areas above the axis add to the total, and the area below the x axis subtract from the total.

Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval, then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width.

The next significant advances in integral calculus did not begin to appear until the 16th century.

She is credited with writing the first book discussing both differential and integral calculus and was an honorary member of the faculty at the University of Bologna.

where again e is the eccentricity and where the function is the complete elliptic integral of the second kind.

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

Modern mathematics defines an " elliptic integral " as any function which can be expressed in the form

However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms ( i. e. the elliptic integrals of the first, second and third kind ).

Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz – Christoffel mapping.

These arguments are expressed in a variety of different but equivalent ways ( they give the same elliptic integral ).

The incomplete elliptic integral of the first kind is defined as

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

For example, some references, and Wolfram's Mathematica software, define the complete elliptic integral of the first kind in terms of the parameter m, instead of the elliptic modulus k.

The incomplete elliptic integral of the second kind in trigonometric form is

The incomplete elliptic integral of the third kind is

The complete elliptic integral of the third kind can be defined as

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic,

If and then the limit is where is the complete elliptic integral of the first kind

where is the complete elliptic integral of the second kind:

where K ( t ) is a complete elliptic integral of the first kind.

In fact, an elliptic function must have at least two poles ( counting multiplicity ) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of any simple poles must cancel.

which is an implicit solution involving an elliptic integral.

This requires an elliptic integral to find, given the polar and equatorial radii:

For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the ( complex ) elliptic integral ( Brent, 1976 ).

The,, and are complete elliptic integral of first, second, and third kind.

The incomplete elliptic integral of the first kind is defined as,

* An integral equation is an equation involving integrals.

Clenshaw – Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler – Maclaurin approach is very accurate ( in that particular case the Euler – Maclaurin formula takes the form of a discrete cosine transform ).

Abramowitz and Stegun substitute the integral of the first kind,, for the argument in their definition of the integrals of the second and third kinds, unless this argument is followed by a backslash: i. e. for.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.

In mathematics, the Cauchy integral theorem ( also known as the Cauchy – Goursat theorem ) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane.

Such a combination is called a closed chain and one defines integral along the chain as the linear combination of integrals over individual paths.

If the boundary is piecewise smooth, then we interpret the integral as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary.

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals ; given a single function to integrate, the typical strategy is to carefully separate it into a product of two functions u ( x ) v ( x ) such that the integral produced by the integration by parts formula is easier to evaluate than the original one.

In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals.

one approach is to phrase the multiple integral as repeated one-dimensional integrals by appealing to Fubini's theorem.

The dominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itō form.

A compilation of a list of integrals ( Integraltafeln ) and techniques of integral calculus was published by the German mathematician Meyer Hirsch in 1810.

Nonetheless their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral

In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals.

and after routine transformations, the integral along the bisector of the first quadrant can be related to the limit of the Fresnel integrals.

However, similar ideas have been used before and go back at least to the introduction of the Riemann – Stieltjes integral which unifies sums and integrals.

For multi-loop integrals that will depend on several variables we can make a change of variables to polar coordinates and then replace the integral over the angles by a sum so we have only a divergent integral, that will depend on the modulus and then we can apply the zeta regularization algorithm, the main idea for multi-loop integrals is to replace the factor after a change to hyperspherical coordinates so the UV overlapping divergences are encoded in variable r. In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as so the multi-loop integral will converge for big enough's ' using the Zeta regularization we can analytic continue the variable's ' to the physical limit where s = 0 and then regularize any UV integral.