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.
A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meier Hirsch [de] (also spelled Meyer Hirsch) in 1810.
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.
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.
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.
More detail may be found on the following pages for the lists of integrals: Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll's (GR) 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).
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.
Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration.
These formulas only state in another form the assertions in the table of derivatives.
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:[2]
where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a piecewise constant function): If the function f does not have any continuous antiderivative 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)) ∫ f(x) dx is an antiderivative 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 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 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.
If the function f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral: