Symbolic Integration

In mathematical analysis, symbolic integration is the finding of a primitive or indefinite integral of a given function f ( x ), that is, a search for a differentiable function F ( x ) such that

{\frac {dF}{dx}}=f(x).

{\ displaystyle {\ frac {dF} {dx}} = f (x).}

{\ displaystyle {\ frac {dF} {dx}} = f (x).}

Designation:

F(x)=\int f(x)\,dx.

{\ displaystyle F (x) = \ int f (x) \, dx.}

{\ displaystyle F (x) = \ int f (x) \, dx.}

The term symbolic is used to distinguish from numerical integration , in which the specific value of a certain integral is calculated $\textstyle F(x)=\int \limits _{a}^{b}f(x)\,dx$ ${\ displaystyle \ textstyle F (x) = \ int \ limits _ {a} ^ {b} f (x) \, dx}$ ${\ displaystyle \ textstyle F (x) = \ int \ limits _ {a} ^ {b} f (x) \, dx}$ by the values of f ( x ).

Both tasks were of great theoretical and practical importance long before the era of digital computers, but now their research is carried out in the field of computer science , as computer algebra systems are created and developed.

Searching for a derivative is a simple process for which it is easy to define an algorithm. The inverse problem is much more complicated, often the integral of an elementary function is not representable in a closed form (a combination of a finite number of elementary functions). See antiderivative .

A procedure called the Rish algorithm is able to determine if an integral exists and find it for many classes of functions. This algorithm continues to improve.

Examples

\int x^{2}\,dx={\frac {x^{3}}{3}}+C

{\ displaystyle \ int x ^ {2} \, dx = {\ frac {x ^ {3}} {3}} + C}

symbolic result (indefinite integral), C - integration constant;

\int \limits _{-1}^{1}x^{2}\,dx={\frac {2}{3}}

{\ displaystyle \ int \ limits _ {- 1} ^ {1} x ^ {2} \, dx = {\ frac {2} {3}}}

symbolic result (definite integral);

\int \limits _{-1}^{1}x^{2}\,dx\approx 0{,}6667

{\ displaystyle \ int \ limits _ {- 1} ^ {1} x ^ {2} \, dx \ approx 0 {,} 6667}

numerical result for this example.

Directories

Symbolic Integration 1 (transcendental functions) by Manuel Bronstein, 1997 by Springer-Verlag, ISBN 3-540-60521-5
Joel Moses , Symbolic integration: the stormy decade, Proceedings of the second ACM symposium on Symbolic and algebraic manipulation, p. 427-440, March 23-25, 1971, Los Angeles, California, United States
KO Geddes and TC Scott, Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms , Proceedings of the 1989 Computers and Mathematics conference, (held at MIT June 12, 1989), edited by E. Kaltofen and SM Watt, Springer-Verlag, New York, (1989), pp. 192-201. [one]
KO Geddes, ML Glasser, RA Moore and TC Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions , AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149–165, [2]

Links

Bhatt, Bhuvanesh. Risch Algorithm on the Wolfram MathWorld website.
Wolfram Integrator - Calculating Integrals Online Using Mathematica
Mathematical Assistant on Web - online symbolic computing
Online Integral Calculator
Online integral calculator with a detailed step-by-step solution in Russian

Symbolic Integration

Examples

See also

Directories

Links

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