IntegrateChangeVariables

IntegrateChangeVariables[integral,u,trans]

changes the variable in integral to the new variable u using the transformation trans.

IntegrateChangeVariables[integral,{u,v,…},trans]

changes the variables to the new variables u, v, ….

Details and Options

  • IntegrateChangeVariables is also known as integration by substitution, u-substitution and reverse chain rule.
  • A change of variables is often used in calculus to simplify an integral by applying a suitable substitution to it or by representing it in another coordinate system to exploit the symmetry in the problem.
  • IntegrateChangeVariables can be used to perform a change of variables for indefinite integrals, definite integrals, multiple integrals and integrals over geometric regions.
  • The change of variables is performed using the change of variables formula
  • on an interval or
  • over a region where denotes the Jacobian of the transformation on .
  • The possible forms for integral are the forms supported by Integrate:
  • Integrate[f[x],x]indefinite univariate integral
    Integrate[f[x],{x,a,b}]definite univariate integral
    Integrate[f[x,y,…],x,y,…]indefinite multivariate integral
    Integrate[f[x,y,…],{x,a,b},{y,c,d},…]definite multivariate integral
    Integrate[f[x,y,…],{x,y,…}∈reg]definite multivariate integral over a region
  • Either an unevaluated Integrate[…] or Inactive[Integrate][…] can be used. It is important that the integral does not evaluate, so the safe method is to use Inactive[Integrate][…] which can be produced through Inactivate[integral,Integrate].
  • IntegrateChangeVariables returns the result in the form Inactive[Integrate][…]. Use Activate to evaluate the integral in the new coordinates. »
  • The transformation trans can have the forms:
  • t==ϕ[x]replace ϕ[x] by t
    {u==ϕ[x,y,…],v=ψ[x,y,…],…}replace ϕ[x,y,…] by u and ψ[x,y,…] by v, etc.
    chart1chart2named coordinate systems from CoordinateChartData
  • The transformation is assumed to be differentiable on its domain of definition.
  • When using named coordinate systems, the transformation can be entered in any form accepted by CoordinateTransformData, including {oldsys,metric,dim}{newsys,metric,dim}, {oldsysnewsys,metric,dim} and the various more abbreviated forms.
  • Restrictions on the domains of the variables and parameters in the integral can be specified using Assumptions.

Examples

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Basic Examples  (4)

Apply the change of variables to an indefinite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Create an inactive multiple integral:

Apply a change of variables to the multiple integral:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to an approximation of a multiple integral:

Evaluate the result:

Compare the result with the original approximation of the multiple integral:

Scope  (21)

Indefinite Integrals  (5)

Apply the change of variables to an indefinite integral:

The transformation can be given with the old variables in terms of the new ones, :

Evaluate the result and substitute back to the original variables:

Compare the result with the original integral:

Apply the change of variables to an indefinite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to an indefinite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables and to an indefinite multiple integral:

Evaluate the result:

Compare the result with the original integral:

Change an indefinite integral from Cartesian to planar parabolic coordinates:

Definite Integrals  (6)

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Apply the change of variables to a definite integral:

Evaluate the result:

Compare the result with the original integral:

Multiple Integrals  (8)

Apply a change of variables to a multiple integral:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to a multiple integral:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to a multiple integral:

Evaluate the result:

The numerical value of the above result agrees with the result returned by NIntegrate:

Apply a change of variables to a multiple integral:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to a multiple integral:

Evaluate the result:

Apply a change of variables to a multiple integral:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to a multiple integral:

Evaluate the result:

Apply a change of variables to a multiple integral:

Evaluate the result:

Compare the result with the original integral:

Integrals over Regions  (2)

Apply the change of variables and to a multiple integral over a region:

Evaluate the result:

Compare the result with the original integral:

Apply a change of variables to a multiple integral over a region:

Evaluate the result:

Compare the result with the original integral:

Applications  (4)

Compute the area of the annulus:

The area of the annulus could also be represented by the following integral, agreeing with the result above:

Evaluate the result:

Compute the area of the following region:

The area of the region is represented by the following integral:

The region is transformed into the following square with the transformation and :

Evaluate the result:

Attempt to compute the following definite integral; it takes a long time and only partially evaluates:

Changing to polar coordinates gives a much simpler integral to evaluate:

The numerical value of the above result agrees with the result returned by NIntegrate:

Attempt to compute the following definite integral; it takes a long time and only partially evaluates:

Changing to polar coordinates gives a much simpler integral to evaluate:

The numerical value of the above result agrees with the result returned by NIntegrate:

Properties & Relations  (2)

A result always has an inactive head, irrespective of the form of input:

Use Activate to evaluate the integral:

IntegrateChangeVariables uses information from both CoordinateChartData and CoordinateTransformData:

The foregoing used both the mapping and the coordinate ranges to give as simple a result as possible:

Wolfram Research (2022), IntegrateChangeVariables, Wolfram Language function, https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html.

Text

Wolfram Research (2022), IntegrateChangeVariables, Wolfram Language function, https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html.

CMS

Wolfram Language. 2022. "IntegrateChangeVariables." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html.

APA

Wolfram Language. (2022). IntegrateChangeVariables. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html

BibTeX

@misc{reference.wolfram_2024_integratechangevariables, author="Wolfram Research", title="{IntegrateChangeVariables}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}", note=[Accessed: 24-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_integratechangevariables, organization={Wolfram Research}, title={IntegrateChangeVariables}, year={2022}, url={https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}, note=[Accessed: 24-November-2024 ]}