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,y,}reg]
integrates over the geometric region reg.

Details and OptionsDetails and Options

  • Integrate[f,x] can be entered as .
  • can be entered as EscintEsc or \[Integral].
  • is not an ordinary ; it is entered as EscddEsc or \[DifferentialD].
  • Integrate[f,{x,y,}reg] can be entered as .
  • Integrate[f,{x,xmin,xmax}] can be entered with as a subscript and 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 a construct such as or any expression whose head is not a mathematical function.
  • 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:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    PrincipalValueFalsewhether to find Cauchy principal values
  • 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 .

ExamplesExamplesopen allclose all

Basic Examples  (7)Basic Examples  (7)

Indefinite integral:

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Definite integral:

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Use EscintEsc to enter and EscddEsc to enter :

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Use Ctrl+_ to enter the lower limit, then Ctrl+% for the upper limit:

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Multiple integral with x integration last:

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Integrate over geometric regions, in this case performing a surface integral:

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Enter the same integral using Ctrl+$ for the region and EscelEsc for :

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Integrals that may not converge are by default returned as conditionals:

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Introduced in 1988
(1.0)
| Updated in 2014
(10.0)