Re: Explanation of this hypothesis of SUBSTITUTION RULE requested


Arturo Magidin wrote:
In article <1140384677.940196.196730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<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.

The conditions are perhaps stronger than might be strictly necessary,
but they do prevent any number of problems.

Suppose, for example, that the function g(x) is differentiable, but
whose range is not an interval; for example, g(x) = sec(x). Then you
could run into all sorts of problems. If we let f(x) = 1/x, then

f(g(x))g'(x) = cos(x)*sec(x)*tan(x) = tan(x), so on the left side we
have the integral of the tangent. If we drop all the hypothesis, then
we could assert that

integral(tan(x))dx = integral(1/u)du

which fails, for example, if we take the first integral over the range
0 to pi (where the integral does not exist), and the right side on the
corresponding "interval", which would be tan(0)=0 to tan(pi)=0.

this should be sec(0) to
sec(pi)

since u = g(x) = sec(x)

~~Aitken

.

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