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IntegrateChangeVariables
✖
IntegrateChangeVariables

[Experimental]

✖
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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Apply the change of variables to an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ee07ne
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-coc5io
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ctvj1j
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-oemqoo
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-i2ys4g
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bqgv2l
Out[3]=3

Create an inactive multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-s8q6oq
Out[1]=1

Apply a change of variables to the multiple integral:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bpoz9v
Out[2]=2

Evaluate the result:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bkhh3s
Out[3]=3

Compare the result with the original integral:

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b1vcn0
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b9j9pb
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-dbk09d
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fmabqy
Out[3]=3

Scope  (21)Survey of the scope of standard use cases

Indefinite Integrals  (5)

Apply the change of variables to an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bhgci4
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-e8x90c
Out[2]=2

Evaluate the result and substitute back to the original variables:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ehptv
Out[3]=3

Compare the result with the original integral:

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ison9
Out[4]=4

Apply the change of variables to an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ehfu5d
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ubyzh
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-hxmiyv
Out[3]=3

Apply the change of variables to an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-igrxh
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-f6n9w6
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-099ja
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-hsn0cm
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-c8hbqg
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-urms0
Out[3]=3

Change an indefinite integral from Cartesian to planar parabolic coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-mvfhkn
Out[1]=1

Definite Integrals  (6)

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-mxst0y
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-e9c3jh
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cdohg5
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fsq1gj
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bn872x
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-zt5q2
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-hcfrtw
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ftv4ss
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-e9j86f
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-91fd8
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fvzl09
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-c1ep9n
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b1f8e8
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cb5gfx
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ho87r6
Out[3]=3

Apply the change of variables to a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-je8j4c
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-0za5o
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-hyg05u
Out[3]=3

Multiple Integrals  (8)

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-e3v4i
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-js5hpo
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bkqqhp
Out[3]=3

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cpohxm
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bbssz9
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-m4tfxb
Out[3]=3

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-gf5g1e
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b0nsrq
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bwjvxl
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-brmfq1
Out[4]=4

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-kpe7d
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-d6wzl4
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bvpmy9
Out[3]=3

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ci0oca
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-eh57j
Out[2]=2

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-6q2xv
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-foevcr
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b3mnsc
Out[3]=3

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-n1avom
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-icklnb
Out[2]=2

Apply a change of variables to a multiple integral:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fs8f7z
Out[1]=1

Evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ccyxa6
Out[2]=2

Compare the result with the original integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-bls3yk
Out[3]=3

Integrals over Regions  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-6wjly
In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-iz87k
Out[2]=2

Evaluate the result:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-o3dby9
Out[3]=3

Compare the result with the original integral:

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-doaxg
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fe7lx6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-iehjal
Out[2]=2

Evaluate the result:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-b3zf96
Out[3]=3

Compare the result with the original integral:

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-so2f7
Out[4]=4

Applications  (4)Sample problems that can be solved with this function

Compute the area of the annulus:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-fz1nh9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-iu0g59
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ddsuez
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-g92e7x
Out[4]=4

Evaluate the result:

In[5]:=5
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ch7c6z
Out[5]=5

Compute the area of the following region:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-dhbty
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-4b44o
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-dn3vf0
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-f5k3q
Out[4]=4

Evaluate the result:

In[5]:=5
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cvejrj
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ivank
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cs0br0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-dfxbd
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-qh1rm
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bh1fnvnemssdy-co2z6o
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-dpk7f
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-iw6hi2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ivsxse
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bh1fnvnemssdy-g387ud
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bh1fnvnemssdy-cbt6f8
Out[5]=5

Properties & Relations  (2)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-d4aj5d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-1ybics
Out[2]=2

Use Activate to evaluate the integral:

In[3]:=3
✖
https://wolfram.com/xid/0bh1fnvnemssdy-853jhb
Out[3]=3

IntegrateChangeVariables uses information from both CoordinateChartData and CoordinateTransformData:

In[1]:=1
✖
https://wolfram.com/xid/0bh1fnvnemssdy-ry8qso
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bh1fnvnemssdy-xw6qoj
Out[2]=2
Wolfram Research (2022), IntegrateChangeVariables, Wolfram Language function, https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html.
✖
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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_integratechangevariables, author="Wolfram Research", title="{IntegrateChangeVariables}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}", note=[Accessed: 24-March-2025 ]}

✖
@misc{reference.wolfram_2025_integratechangevariables, author="Wolfram Research", title="{IntegrateChangeVariables}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}", note=[Accessed: 24-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_integratechangevariables, organization={Wolfram Research}, title={IntegrateChangeVariables}, year={2022}, url={https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}, note=[Accessed: 24-March-2025 ]}

✖
@online{reference.wolfram_2025_integratechangevariables, organization={Wolfram Research}, title={IntegrateChangeVariables}, year={2022}, url={https://reference.wolfram.com/language/ref/IntegrateChangeVariables.html}, note=[Accessed: 24-March-2025 ]}