Theorem:
integral_mult
  not (?b contains ?bdv) => Integral ?a * ?b D ?bdv = ?a * Integral ?b D ?bdv

Proof of the theorem

Explanations:

The precondition not (?b contains ?bdv) inhibits wrong applications of the rule as for instance

Integral x* x D x = Integral x D x * Integral x D x ((1))

This is wrong as you easily can verify: integrating the left hand side in (1) gives

Integral x* x D x = Integral x^2 D x = x^3/3 + c ((2))

while integrating the right hand sidein (1) gives

Integral x D x * Integral x D x = x^2/2 * x^2/2 + c = x^4/4 + c ((3))

Thus integrating the both sides in (1) gives two different results (2) and (3) which should not happen (as long as we want to perserve the meaning of '=' and the Integral to be welldefined).


(c) Richard Lang 2003 (math-author)
(c) isac-team 2006 (course-designer)