ibpdv is used to search the primitive of an expression written
as u(x).v′(x).
ibpdv takes two arguments :
ibpdv returns :
Hence, ibpdv returns the terms computed in an integration by parts,
with the possibility of doing several ibpdv succesively.
When the answer of ibpdv(u(x)*v’(x),v(x)) is computed, to obtain a
primitive of u(x).v′(x), it remains to
compute the integral of the second term of this answer and then, to sum this
integral with the first term of this answer : to do this, just use
ibpdv command with the answer as first argument and
a new v(x) (or 0 to terminate the integration) as second argument.
Input :
Output :
then
Output :
Remark
When the first argument of ibpdv is a list of two elements, ibpdv
works only on the last element of this list and adds the integrated term to
the first element of this list.
(therefore it is possible to do several ibpdv successively).
For example :
ibpdv((log(x))^
2,x) = [x*(log(x))^
2,-(2*log(x))]
it remains to integrate -(2*log(x)), the input :
ibpdv(ans(),x) or input :
ibpdv([x*(log(x))^
2,-(2*log(x))],x)
Output :
[x*(log(x))^
2+x*(-(2*log(x))),2]
and it remains to integrate 2, hence input ibpdv(ans(),0) or
ibpdv([x*(log(x))^
2+x*(-(2*log(x))),2],0).
Output :
x*(log(x))^
2+x*(-(2*log(x)))+2*x
ibpu is used to search the primitive of an expression written as u(x).v′(x) ibpu takes two arguments :
ibpu returns :
Hence, ibpu returns the terms computed in an integration by parts,
with the possibility of doing several ibpu succesively.
When the answer of ibpu(u(x)*v’(x),u(x)) is computed, to obtain a
primitive of u(x).v′(x), it remains to
compute the integral of the second term of this answer and then, to sum this
integral with the first term of this answer : to do this, just use
ibpu command with the answer as first argument and
a new u(x) (or 0 to terminate the integration) as second argument.
Input :
Output :
then
Output :
Remark
When the first argument of ibpu is a list of two elements, ibpu
works only on the last element of this list and adds the integrated term to
the first element of this list.
(therefore it is possible to do several ibpu successively).
For example :
ibpu((log(x))^
2,log(x)) = [x*(log(x))^
2,-(2*log(x))]
it remains to integrate -(2*log(x)), hence input :
ibpu(ans(),log(x))
or input :
ibpu([x*(log(x))^
2,-(2*log(x))],log(x))
Output :
[x*(log(x))^
2+x*(-(2*log(x))),2]
it remains to integrate 2, hence input :
ibpu(ans(),0) or input :
ibpu([x*(log(x))^
2+x*(-(2*log(x))),2],0).
Output :
x*(log(x))^
2+x*(-(2*log(x)))+2*x