Re: Explanation of this hypothesis of SUBSTITUTION RULE requested

Arturo Magidin wrote:
In article <kx8Kf.3041$%14.96328@xxxxxxxxxxxxxxxxxxxxx>,
True Raptor <CB4ever@xxxxxxxxxxxxxxx> wrote:

deniz.bahar@xxxxxxxxx wrote:

My question deals with this following RULE/THEOREM:

The Substitution Rule: If u=g(x) is a differentiable function whose
range is an inteval on which f is continuous, then

integral[ f (g(x)) g' (x) ] = integral[ f (u) du ]


I can't understand why the condition "whose range is an interval on
which f is continuous" is needed. What are some cases that could
break this, and not allow us to use substitution (yet look on the
surface as a good place to use substit.) ?


If g is defined for some values for which f is not continuous, then the integral does not exist.


May fail to exist.


Integration requires continuity.


No, it does not require continuity. Continuity guarantees the
existence of the integral, but is not required. A step function, for
example, is easily shown to be integrable even if it is not exontinuous.


Spivak's calculus (last edition, pg 365) says that f and g' have to be continuous to use the substitution rule:

def. integral from g(a) to g(b) of: f

=

def. integral from a to b of: (fog) g'


For indefinite integration, Spivak (pg 367) poses no such hypothesis/restriction for one to use substition to change the original integral to a form with x's into one with u's (using u=g(x)), then convert it back to an expression with x after antidifferentiation has been done.

This suggests that the caveats associated with u=g(x) and it's relationship to f only apply to definite integration.

CB4Ever

.

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