# Lists of integrals

## Information about Lists of integrals

See the following pages for lists of integrals: Also see table of integrals for the most common integral functions.

## Historical developments

A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meyer Hirsch in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Rhyzik. In Gradshteyn and Rhyzik, integrals originating from the book by de Bierens are denoted by BI.

## Other lists of integrals

Gradshteyn and Ryzhik contains a large collection of results. Other useful resources include the CRC Standard Mathematical Tables and Formulae and Abramowitz and Stegun. A&S contains many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. There are several web sites which have tables of integrals and integrals on demand.

## References

### Historical

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
secant joining the endpoints of the interval [ab] is parallel to the tangent at c.]] In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions.

It may be stated thus:

or in the Leibniz notation thus:

In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.

If the function one wishes to differentiate, , can be written as

In calculus, the chain rule is a formula for the derivative of the composite of two functions.

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
In differential calculus, related rates problems involve finding the rate at which a quantity is changing by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.
The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, with respect to x, from the real numbers, and c is a real number.
INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation.
Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks (of varying radius) of infinitesimal thickness.
Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

It makes use of the so-called "representative cylinder".
In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians.
In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing the radical expressions:

INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
partial fractions in integration, by decomposing the rational function into a sum of functions of the form:
.

The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, please see table of integrals and list of integrals.
The following is a list of integrals (antiderivative functions) of trigonometric functions. For a complete list of Integral functions, see table of integrals and list of integrals. See also: trigonometric integral

The constant c is assumed to be nonzero.
Substitution or other forms of algebraic manipulation are used to arrive at integrals listed in the table of integrals.