Symbolic integration

As algebraic or symbolic integration or quadrature is called in mathematics the calculation of integrals by exact term transformations, in contrast to the approximate numerical quadrature.

The algebraic integration is one of the most important applications of computer algebra systems. These programs are implemented to determine a primitive functions. The main rules here are the substitution rule and integration by parts. However, one quickly arrives at these techniques to the limits of their applicability. Thus, the function has no closed-form representation of its common function. For these cases, there are also the techniques of Fourier transform and the residue theorem, which are also dominated by modern computer algebra systems. In addition, computer algebra systems use so-called error - in functions such as the Gaussian error function in the formation of primitive functions which have no closed form.

There is a method called Risch algorithm, which can determine terms for many classes of integrands, whether integral exists, and this is then determined. Such algorithms are still evolving, because of the Risch algorithm is restricted to indefinite integrals. However, the vast majority of interest to physicists, theoretical chemists and engineers integrals are definite integrals, often with respect to the Laplace transform, Fourier transform or the Mellin transform. An alternative to the Risch algorithm uses a combination of computer algebra systems and pattern recognition as well as the knowledge of special functions, in particular the incomplete gamma function. Although this approach is more heuristic than algorithmic, but it is an effective method for the calculation of definite integrals, particularly those that occur in the practice of engineering. This method was first implemented by the developers of the computer algebra system Maple, and later adopted by systems such as Mathematica, MuPAD, and others.

Example

Using the polynomial function is a simple example is given. so is

The symbolic result of the indefinite integral, wherein an integration constant. For the definite integral

Is the symbolic value and the numeric is 0.6666 .... The number of decimal places is infinite.