Integral panel

Integral panel by
Meier Hirsch , 1810

Integral panel by
Ferdinand Minding, 1849

Gradshteyn-Ryzhik
7th edition 2007

An integral table is a printed work or a file in which the integrals of numerous functions, usually the functions of a variable, are compiled in tabular form. The significance of this compilation is that there is no standard procedure or algorithm for determining integrals and this can be extremely difficult in a specific case. Integral tables are therefore always used when an exact evaluation of an integral is required in closed form , for example when a numerical calculation of the integral does not continue.

In the meantime, however, computer algebra provides very effective calculation programs with which many (but never all) integrals can also be calculated symbolically in closed form with the help of elementary and special functions .

The special thing about an integral table compared to a calculation using computer algebra is its quality as a reference work. An integral table does more because it can also be read in the opposite direction: it can be used to find integral representations of functions or mathematical constants .

An integral table is usually divided into two parts:

• A list of indefinite integrals , i.e. of antiderivatives. In order to capture as many functions as possible, parameters are used. In some cases different antiderivatives result, which of course only differ by one additive constant:

${\displaystyle \int \sin x\cdot \cos x\,\mathrm {d} x={\tfrac {1}{2}}\sin ^{2}x+c_{1}=-{\tfrac {1}{2}}\cos ^{2}x+c_{2}}$

However, this is by no means always easy to see.

• A list of definite integrals . This mainly includes cases where an indefinite integral cannot be given in closed form. For certain important or special integration limits (often multiples of these, or ) the exact value of the integral can nonetheless be determined, and this is listed here. Here, too, parameters are used in order to capture as many integrals as possible. In addition to mathematical constants, special functions also appear in the evaluation of the integrals. ${\displaystyle 0,1,\pi ,e,}$ ${\displaystyle -\infty ,\infty }$

The classification and arrangement of the integrals are problematic. When searching, it must be taken into account that a specific function can possibly be classified in different groups. It must also be noted that the integrals of two similar output functions are often no longer related at all and unexpectedly special functions occur in the evaluation, since the integration leads to an expansion of given function classes.

In many cases, tables of finite and infinite series are also included.

The creation of tables of definite integrals is far more difficult than that of indefinite integrals (both for calculation and for classification and arrangement). Therefore, many integral tables contain only a small selection of certain integrals. For the latter, however, there are very extensive special boards that have been and are supplemented by widely distributed magazine articles. There have been repeated efforts to bring these together in even more extensive works and to supplement them with sources and evidence. George Boros and Victor Moll write in the foreword to Irresistible Integrals :

"It took a short time to realize that this task was monumental."

"It only took a short time to realize that this task was an enormous effort."

Computer algebra offers a certain alternative to integral tables. Alexander Apelblat noted in his foreword in 1982:

"As yet, the necessity for rapid and convenient evaluation of integrals has not been eliminated by computers."

"Until now, the need to evaluate integrals quickly and easily has not been eliminated by computers."

In the meantime, powerful computer algebra systems such as Mathematica or Maple can integrate more and, above all, more complex functions than traditional printed boards, but by no means all of the previously tabulated ones. In particular, very special, definite and improper integrals cause difficulties or are calculated incorrectly.

history

Although Leibniz , the brothers Jakob I and Johann I Bernoulli and Euler calculated many integrals and published them in their work, independent integral tables did not appear until the beginning of the 19th century:

• 1810: Meier Hirsch : Integraltafeln, or collection of integral formulas . Duncker & Humblot, Berlin. (303 p.)
• 1849: Ferdinand Minding : Collection of integral panels for use in teaching at the Königl. General Building School and the Königl. Commercial Institute . Reimarus, Berlin. (186 p.)
• 1943: Gradshteyn-Ryzhik : Table of Integrals, Series and Products. 8th edition 2014. (1171 pp.)
• 1944: Wolfgang Gröbner , Nikolaus Hofreiter : Integraltafel I: indefinite integrals, II: certain integrals , Springer-Verlag, 5th edition, vol. 1: 1975 (166 p.), Vol. 2: 1973 (204 p.)
• 1981: AP Prudnikov , Yuri A. Brychkov , OI Marichev : Integrals and series (5 volumes). 2nd edition Moscow 2003. (Vol. 1: 631 pp.)
• 1982: Alexander Apelblat : Table of definite and infinite integrals , Elsevier, 1983. (457 pp.)
• 1996: Alexander Apelblat: Collection of definite, infinite and indefinite integrals and infinite series , Harri Deutsch, 1996. (286 p., Is a supplement to 1983)

With very few exceptions, Hirsch only contains indefinite integrals, Minding already contains 24 pages with definite integrals.

See also

• Some important indefinite integrals can be found in the table of derivative and antiderivatives .
• There is a quite extensive integral table at Wikibooks in the mathematical formula collection.

Web links

• Integral calculator at WolframAlpha

literature

• George Boros, Victor H. Moll: Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals , Cambridge University Press, 2004, ISBN 0-521-79636-9 .