Integration by parts : ibpdv et ibpu

### 2.18.4 Integration by parts : `ibpdv` et `ibpu`

#### `ibpdv`

`ibpdv` is used to search the primitive of an expression written as u(x).v′(x).
`ibpdv` takes two arguments :

- an expression u(x).v′(x) and v(x) (or a list of two expressions [F(x), u(x)*v′(x)] and v(x)),
- or an expression g(x) and 0 (or a list of two expressions [F(x), g(x)] and 0).

`ibpdv` returns :

- if v(x) ≠ 0, the list [u(x).v(x),−v(x).u′(x)] (or [F(x)+u(x).v(x),−v(x).u′(x)]),
- if the second argument is zero, a primitive of the first argument g(x) (or F(x)+a primitive of g(x)) :
  hence, `ibpdv(g(x),0)` returns a primitive `G(x)` of `g(x)` or
  `ibpdv([F(x),g(x)],0)` returns `F(x)+G(x)` where `diff(G(x))=g(x)`.

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 :

`ibpdv(ln(x),x)`

Output :

`[x.ln(x),-1]`

then

`ibpdv([x.ln(x),-1],0)`

Output :

`-x+x.ln(x)`

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`

`ibpu` is used to search the primitive of an expression written as u(x).v′(x) `ibpu` takes two arguments :

- an expression u(x).v′(x) and u(x) (or a list of two expressions [F(x), u(x)*v′(x)] and u(x)),
- an expression g(x) and 0 (or a list of two expressions [F(x) g(x)] and 0).

`ibpu` returns :

- if u(x) ≠ 0, the list [u(x).v(x),−v(x).u′(x)] (or returns the list [F(x)+u(x).v(x),−v(x).u′(x)]),
- if the second argument is zero, a primitive of the first argument g(x) (or F(x)+a primitive of g(x)):
  `ibpu(g(x),0)` returns `G(x)` where `diff(G(x))=g(x)` or
  `ibpu([F(x),g(x)],0)` returns `F(x)+G(x)` where `diff(G(x))=g(x)`.

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 :

`ibpu(ln(x),ln(x))`

Output :

`[x.ln(x),-1]`

then

`ibpu([x.ln(x),-1],0)`

Output :

`-x+x.ln(x)`

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`