Re: Substitutions in Indefinite Ingegration With a Single Variable

In article <1179948276.219396.27150@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
BrandonFromFlorida <BrandonShw@xxxxxxx> wrote:
On May 23, 12:34 pm, magi...@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <1179914905.640343.207...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,


....If you want to do the integral of 1/(1-x^2), then you do partial
fractions, not a substitution.

I know that's a partial fractions problem, and I've known it for about
35 years since I was in college. In fact it's one of the most classic
partial fractions integrations. I didn't ask you how to integrate
it. My question was, if a hypothetical student asks if the
substitution x = sin(theta), which worked in 1/SQRT(1-x^2) can be
applied to 1/(1-x^2), is it valid for me to simply answer "No, because
the function being substituted has a much smaller range than the
integrand?"

Do you remember how you started this thread? You started by saying
that the integrand was "nice" or well-behaved, or some such.

1/(1-x^2) is NOT "nice" and not "well-behaved". Its domain is
disconnected. When you do an indefinite integral, you have implicit
assumptions in there.

You can certainly do a substitution with x=sin(u). HOWEVER, that
substitution will only yield a valid antiderivative in a certain part
of the domain of the function, namely, the connected component
(-1,1).

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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