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 Documentation / Mathematica / Built-in Functions / New in Version 3.0 / Algebraic Computation  /
Integrate

  • Integrate[ f , x ] gives the indefinite integral .
  • Integrate[ f , x , xmin , xmax ] gives the definite integral .
  • Integrate[ f , x , xmin , xmax , y , ymin , ymax ] gives the multiple integral .
  • Integrate[ f , x ] can be entered as f x.
  • can be entered as int or \[Integral].
  • is not an ordinary d; it is entered as dd or \[DifferentialD].
  • Integrate[ f , x , xmin , xmax ] can be entered with xmin as a subscript and xmax as a superscript to .
  • Multiple integrals use a variant of the standard iterator notation. The first variable given corresponds to the outermost integral, and is done last.
  • Integrate can evaluate integrals of rational functions. It can also evaluate integrals that involve exponential, logarithmic, trigonometric and inverse trigonometric functions, so long as the result comes out in terms of the same set of functions.
  • Integrate can give results in terms of many special functions.
  • Integrate carries out some simplifications on integrals it cannot explicitly do.
  • You can get a numerical result by applying N to a definite integral.
  • You can assign values to patterns involving Integrate to give results for new classes of integrals.
  • The integration variable can be any expression. However, Integrate uses only its literal form. The object , for example, is not converted to .
  • For indefinite integrals, Integrate tries to find results that are correct for almost all values of parameters.
  • For definite integrals, the following options can be given:
  • Integrate can evaluate essentially all indefinite integrals and most definite integrals listed in standard books of tables.
  • In StandardForm, Integrate[ f , x ] is output as f x.
  • See the Mathematica book: Section 1.4.4, Section 1.5.3Section 3.5.6.
  • See also Implementation NotesA.9.55.11MainBookLinkOldButtonDataA.9.55.11.
  • See also: NIntegrate, DSolve, Sum.

    Further Examples

    Indefinite integrals
    Here are three indefinite integrals that are evaluated by the (extended) Risch algorithm.

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    Here are two indefinite integrals that are evaluated by special table lookup rules.

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    Definite integrals
    Here is a definite integral.

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    Options: Assumptions and GenerateConditions
    When an integrand depends on a parameter, the indefinite integral should be considered valid for "generic" values of the parameter. For certain values the reported integral may be meaningless, as is the case here for .

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    By contrast, when you ask for a definite integral, Mathematica tries to return a result that is always valid, if necessary by stating validity conditions.

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    The option Assumptions lets you explicitly state conditions on the parameters. If you are only interested in real values of that are less than 0, the reported result is unconditional.

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    The same unconditional result is returned if you override the default setting of the option GenerateConditions. Unless you know beforehand under what conditions the result is valid, this usage can lead to nonsense.

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    This integral is a definition of the Beta function.

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    Here is a more complicated example. For , the reported indefinite integral is not valid; the true integral is Log[x], since the integrand reduces to .

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    Here is the result when you ask for a definite integral.

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    Even for a definite integral, you should use results with caution if they depend on a parameter. In this example, the result is not valid if is such that there is a singularity in the segment (that is, if is real and at least ).

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    Pi x
    Integrate::idiv: Integral of Tan[----] does not converge on {0, 1}.
    2

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    Options: PrincipalValue
    Sometimes an integrand has a singularity on the path of integration that causes the integral (in the Riemann sense of an area) to diverge. This is the case for this function, at the point .

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    Sqrt[x]
    Integrate::idiv: Integral of ------- does not converge on {0, 3}.
    -1 + x

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    The integral may nonetheless exist in the sense of the Cauchy principal value. This value is defined by considering the integral obtained by omitting a small interval centered at the singularity, and taking the limit as the interval's length shrinks to zero.

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    Cauchy principal values can also be obtained numerically, using the package "NumericalMath`CauchyPrincipalValue`".

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    Numerical integration
    When a function cannot be integrated symbolically, you can usually obtain a definite integral numerically using NIntegrate.

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    See the Further Examples for NIntegrate.