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. chart1chart2 named 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: 06-August-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: 06-August-2024 ]}