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Integration is the basic operation in integral calculus. While differentiation has easy 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.
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 Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by de Bierens 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 e^{−x2}, whose antiderivative is (up to constants) 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.
More detail may be found on the following pages for the lists of integrals:
Gradshteyn, Ryzhik, Jeffrey, Zwillinger'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 (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms). 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, Bronstein and Semendyayev's Handbook of Mathematics (Springer) and Oxford Users' Guide to Mathematics (Oxford Univ. Press), 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 Wolfram Mathematica Online Integrator.
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.
When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), 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 would 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 −iπ when using a path above the origin and iπ 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:
These rational functions have a non-integrable singularity at 0 for a ≤ −1.
Let f be a function which has at most one root on each interval on which it is defined, and g an antiderivative of f that is zero at each root of f (such an antiderivative exists if and only if the condition on f is satisfied), then
where sgn(x) is the sign function, which takes the values -1, 0, 1 when x is respectively negative, zero or positive. This gives the following formulas (where a≠0):
when ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) \, for some integer n.
when ax \in \left( n\pi, n\pi + \pi \right) \, for some integer n.
If the function f does not has any continuous anti-derivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then \sgn(f(x)) \int f(x)dx is an anti-derivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x)=0. For having a continuous anti-derivative, 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 π, then we get:
Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function
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 f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral:
The change of variable x = e^{-t} gives
and, under this form the result appears in List of integrals of exponential functions#Definite integrals
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