Lists of integrals


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.

Historical development of integrals

A compilation of a list of integrals and techniques of integral calculus was published by the German mathematician in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his Tables d'intégrales définies, supplemented by Supplément aux tables d'intégrales définies in ca. 1864. A new edition was published in 1867 under the title Nouvelles tables d'intégrales définies. 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 Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed form antiderivatives. A simple example of a function without a closed form antiderivative is, whose antiderivative is the error function.
Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.

Lists of integrals

More detail may be found on the following pages for the lists of integrals:
Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, Moll's Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev. More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of Indefinite Integrals, or as chapters in Zwillinger's CRC Standard Mathematical Tables and Formulae or Bronshtein and Semendyayev's Guide Book to Mathematics, Handbook of Mathematics or Users' Guide to Mathematics, and other mathematical handbooks.
Other useful resources include Abramowitz and Stegun and the Bateman Manuscript Project. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
There are several web sites which have tables of integrals and integrals on demand. Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration. Wolfram Research also operates another online service, the .

Integrals of simple functions

C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of antiderivatives.
These formulas only state in another form the assertions in the table of derivatives.

Integrals with a singularity

When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point, then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is not in general necessary. For instance in
there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −i when using a path above the origin and i for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:

Rational functions

The following function has a non-integrable singularity at 0 for :

Exponential functions

Logarithms

Trigonometric functions

Inverse trigonometric functions

Hyperbolic functions

Inverse hyperbolic functions

Products of functions proportional to their second derivatives

Absolute-value functions

Let be a function which has at most one root on each interval on which it is defined, and an antiderivative of that is zero at each root of , then
where is the sign function, which takes the values −1, 0, 1 when is respectively negative, zero or positive. This gives the following formulas :
when for some integer.
when for some integer.
when for some integer.
when for some integer.
If the function does not have any continuous antiderivative which takes the value zero at the zeros of , then is an antiderivative of on every interval on which is not zero, but may be discontinuous at the points where. For having a continuous antiderivative, one has thus to add a well chosen step function. If we also use the fact that the absolute values of sine and cosine are periodic with period, then we get:

Special functions

Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function

Definite integrals lacking closed-form antiderivatives

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
If the function has bounded variation on the interval, then the method of exhaustion provides a formula for the integral:
The "sophomore's dream":
attributed to Johann Bernoulli.
S Calculated

Tables of integrals

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