Symbolic integration

In calculus symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find the differentiable function F(x) such that

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

This is also denoted

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

The term symbolic is used to distinguish this problem from that of numerical integration, where the value of F at a particular input or set of inputs, rather than a general formula for F, is sought.

Both problems were held to be of practical and theoretical importance long before the time of digital computers, but they are now generally considered the domain of computer science, as computers are most often used currently to tackle individual instances.

Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm. The reverse question of finding the integral is much more difficult. Many expressions which are relatively simple do not have integrals that can be expressed in closed form. See antiderivative for more details.

A procedure called the Risch algorithm exists which is capable of determining if an integral exists and returning it if it does, for many classes of expressions. Such algorithms are still being expanded.

However, the Risch algorithm applies to indefinite integrals and most of the integrals of interest to Physicists, theoretical Chemists and Engineers, are definite integrals often related to Laplace transforms, Fourier transforms and Mellin transforms. An alternative to the Risch algorithm involves the combination of a computer algebra system, pattern-matching and the exploitation of special functions, in particular the Incomplete gamma function Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving definite integrals, in particular those encountered by practical engineering applications. This method was pioneered by developers of the Macsyma, Reduce and Axiom computer algebra systems, then later emulated by Maple, Mathematica, MuPAD and other systems. This approach has applications in experimental mathematics.

Example

For example:

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

is a symbolic result for an indefinite integral (here C is a constant of integration),

$\int_{-1}^1 x^2\,dx = \left[\frac{x^3}{3}\right]_{-1}^1= \frac{1^3}{3} - \frac{(-1)^3}{3}=\frac{2}{3}$

is a symbolic result for a definite integral, and

$\int_{-1}^1 x^2\,dx \approx 0.6667$

is a numerical result for the same definite integral.

See also

• Antiderivative
• Definite integral
• Elementary function
• Risch algorithm
• Symbolic computation

References

• Bronstein, Manuel (1997), Symbolic Integration 1 (transcendental functions) (2 ed.), Springer-Verlag, ISBN 3-540-60521-5 
• Moses, Joel (March 23–25, 1971), "Symbolic integration: the stormy decade", Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation (Los Angeles, California): 427–440 

External links

• Bhatt, Bhuvanesh, "Risch Algorithm" from MathWorld.
• Wolfram Integrator — Free online symbolic integration with Mathematica
• Mathematical Assistant on Web — symbolic computations online. Allows to integrate in small steps (with hints for next step (integration by parts, substitution, partial fractions, application of formulas and others), powered by Maxima
• Online integral calculator.