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 Options

  • Integrate[f,x] can be entered as ∫fx.
  • ∫ can be entered as int or \[Integral].
  • is not an ordinary d; it is entered as dd or \[DifferentialD].
  • Integrate[f,{x,y,…}∈reg] can be entered as ∫{x,y,…}∈regf.
  • 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 a construct such as x[i] 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 ∫fx.

Examples

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Basic Examples  (7)

Indefinite integral:

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Definite integral:

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Use int to enter ∫ and dd to enter :

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Use to enter the lower limit, then 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 for the region and el for ∈:

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

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Scope  (36)

Generalizations & Extensions  (3)

Options  (5)

Applications  (6)

Properties & Relations  (10)

Possible Issues  (14)

Neat Examples  (4)

See Also

NIntegrate  DSolve  Sum  LaplaceTransform  FourierTransform  Convolve  D  Derivative  CDF  Expectation  Probability  ArcLength  Area  Volume  MomentOfInertia

Tutorials

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