From theorem 5.9 about the derivative of the composition of two functions, we deduce the method of integration by substitution (i.e. change of variable).
Let
and
be two functions, each of them being differentiable on its own domain. By Theorem 5.9,
is differentiable and we have
. It follows that:
Denote
; then
and Equation 5 can be written in the following form:
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Denote
. We have:
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Now let
; then
, i.e.
. We have:
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We have:
, thus:
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Denote
, thus
and we have:
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Noah Dana-Picard 2007-12-28