IntegrationTools
Parts
perform integration by parts
Calling Sequence
Parameters
Options
Description
Examples
Parts(t, u)
Parts(t, u, v)
Parts(t, u, applytoall)
Parts(t, u, v, applytoall)
t
-
expression containing definite or indefinite integrals
u
u-term
v
v-term
applytoall
If there is more than one integral in the input, the applytoall option will perform integration by parts on each.
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.
withIntegrationTools:
V≔Intexpxsinx,x
V≔∫ⅇxsinxⅆx
PartsV,sinx
ⅇxsinx−∫ⅇxcosxⅆx
PartsV,expx
−ⅇxcosx−∫−ⅇxcosxⅆx
Definite integral.
V≔Intexpxsinx,x=a..b
V≔∫abⅇxsinxⅆx
ⅇbsinb−ⅇasina−∫abⅇxcosxⅆx
−ⅇbcosb+ⅇacosa−∫ab−ⅇxcosxⅆx
Specifying both u and v.
V≔Intfxgx,x=a..b
V≔∫abfxgxⅆx
PartsV,fx
∫gbⅆbfb−∫gaⅆafa−∫ab∫gxⅆxⅆⅆxfxⅆx
PartsV,fx,Gx
Gbfb−Gafa−∫abGxⅆⅆxfxⅆx
Dealing with multiple integrals
U≔Intexpxsinx,x
U≔∫ⅇxsinxⅆx
V≔Intx2sinx,x
V≔∫x2sinxⅆx
W≔valueV
W≔−x2cosx+2cosx+2xsinx
PartsU,sinx
PartsU=W,sinx
ⅇxsinx−∫ⅇxcosxⅆx=−x2cosx+2cosx+2xsinx
PartsU+V,sinx
Error, (in IntegrationTools:-Parts) multiple integrals detected
PartsU+V,sinx,applytoall=true
ⅇxsinx−∫ⅇxcosxⅆx+x3sinx3−∫x3cosx3ⅆx
See Also
Download Help Document