Integral

: This article deals with the concept of an integral in calculus. For other meanings of "integral" see integration and integral (disambiguation).

Computing integrals

The most basic technique for computing integrals of one real variable is based on the Fundamental Theorem of Calculus. It proceeds like this:

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• Choose a function f(x) and an interval .
• Find an antiderivative of f, that is, a function F such that F' =f.
• By the Fundamental Theorem of Calculus, $int\_a^b\; f(x),dx\; =\; F(b)-F(a)$.
• Therefore the value of the integral is F(b)-F(a).

Note that the integral is not actually the antiderivative (the integral is a number), but the fundamental theorem allows us to use antiderivatives to evaluate integrals.

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The difficult step is finding an antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include:

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• Integration by substitution.
• Integration by parts.
• Integration by trigonometric substitution.
• Integration by partial fractions.

Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is residue calculus. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.

Related Topics:
Residue calculus - Parseval's identity - Gaussian integral

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Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.

Related Topics:
Solids of revolution - Disk integration - Shell integration

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Specific results which have been worked out by various techniques are collected in the list of integrals.

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Approximation of definite integrals

Definite integrals may be approximated using several methods of numerical integration. One popular method, called the rectangle method, relies on dividing the region under the function into a series of rectangles and finding the sum. Other well-known methods are the trapezoidal rule and Simpson's rule.

Related Topics:
Numerical integration - Rectangle method - Trapezoidal rule - Simpson's rule

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Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremely time-consuming or computationally-intensive. Approximation, however, is a process which relies only on variable substitution, multiplication, addition, and division. It can be done easily and quickly by modern graphing calculators and computers. Many real-world applications of calculus rely on calculating integrals approximately because of the complexity of formulas and since an exact answer is unnecessary.

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Integrals and computerized algebra systems

Many professionals, educators, and students now use computerized algebra systems to make difficult (or simply tedious) algebra and calculus problems easier. The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation are difficult to formulate.

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One difficulty is that it is not always possible to find "explicit formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no nice function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is exp(−x2). As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function. Unfortunately, functions that have nice antiderivatives are the exception. If one writes a large random expression involving exponentials and polynomials, the odds are almost nil that it will have an antiderivative. (This statement can be made formal, but it is difficult to do so.)

Related Topics:
Explicit formulae - Polynomial - Exponential

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One of the difficulties is to decide what set of functions to use as building blocks for antiderivatives. Usually, we need a set of antiderivatives closed under, say, multiplication and composition. This set of antiderivatives should also include polynomials, perhaps quotients, exponentials, logarithms, sines and cosines. The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it. Extended versions of this algorithm are implemented in Mathematica and the Maple computer algebra system.

Related Topics:
Logarithm - Cosine - Risch-Norman algorithm - Mathematica - Maple computer algebra system

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Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function and so on). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.

Related Topics:
Special functions - Physics - Legendre function - Hypergeometric function - Gamma function

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Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so this advantage is dubious.

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