User:Prime.mover/Proof Structures/Integration by Parts

Integration by Parts

With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:
{{begin-eqn}}
{{eqn | l = u
      | r = ...
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = ...
      | c = link to rule
}}
{{end-eqn}}


and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = ...
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = ...
      | c = link to rule
}}
{{end-eqn}}


Then:
{{begin-eqn}}
{{eqn | l = \int ...
      | r = \int u \rd v
      | c = 
}}
{{eqn | r = \paren {u} \paren {v} - \int \paren {v} \paren {\frac {\d u} {\d x} } \rd x
      | c = [[Integration by Parts]]
}}
{{eqn | r = ...
      | c = etc
}}
{{end-eqn}}
{{qed}}