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IntegrationTools

  

Parts

  

perform integration by parts

 

Calling Sequence

Parameters

Options

Description

Examples

Calling Sequence

Parts(t, u)

Parts(t, u, v)

Parts(t, u, applytoall)

Parts(t, u, v, applytoall)

Parameters

t

-

expression containing definite or indefinite integrals

u

-

u-term

v

-

v-term

Options

• 

applytoall

  

If there is more than one integral in the input, the applytoall option will perform integration by parts on each.

Description

• 

The Parts command performs integration by parts in an integral: ∫uxDvxⅆx=uvvx∫vxDuxⅆx. A similar transformation can be applied to definite integrals as well. By default the Parts command will apply the transformation to t only if it contains a single integral. In case of multiple integrals an error will be thrown. The Parts command can be forced to apply the same transformation to all integrals in t by setting the applytoall option to true.

• 

The first parameter t is the integral.

• 

The second parameter u is the u-term.

• 

The third (optional) parameter v is the v-term. If this term is not specified it will be calculated from the first two parameters.

Examples

withIntegrationTools:

V∫ⅇxsinxⅆx

V:=∫ⅇxsinxⅆx

(1)

PartsV,sinx

ⅇxsinx∫ⅇxcosxⅆx

(2)

PartsV,ⅇx

ⅇxcosx∫ⅇxcosxⅆx

(3)

Definite integral.

V∫abⅇxsinxⅆx

V:=∫abⅇxsinxⅆx

(4)

PartsV,sinx

ⅇbsinbⅇasina∫abⅇxcosxⅆx

(5)

PartsV,ⅇx

ⅇbcosb+ⅇacosa∫abⅇxcosxⅆx

(6)

Specifying both u and v.

V∫abfxgxⅆx

V:=∫abfxgxⅆx

(7)

PartsV,fx

limx→b∫gxⅆxfxlimx→a+∫gxⅆxfx∫ab∫gxⅆxⅆⅆxfxⅆx

(8)

PartsV,fx,Gx

GbfbGafa∫abGxⅆⅆxfxⅆx

(9)

Dealing with multiple integrals

U∫ⅇxsinxⅆx

U:=∫ⅇxsinxⅆx

(10)

V∫x2sinxⅆx

V:=∫x2sinxⅆx

(11)

WvalueV

W:=x2cosx+2cosx+2xsinx

(12)

PartsU,sinx

ⅇxsinx∫ⅇxcosxⅆx

(13)

PartsU=W,sinx

ⅇxsinx∫ⅇxcosxⅆx=x2cosx+2cosx+2xsinx

(14)

PartsU+V,sinx

Error, (in IntegrationTools:-Parts) multiple integrals detected

PartsU+V,sinx,applytoall=true

ⅇxsinx∫ⅇxcosxⅆx+13x3sinx∫13x3cosxⅆx

(15)

See Also

IntegrationTools

 


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