TuringTest
  • TuringTest
Integral\[\int_{-\infty}^{\infty}\frac{dx}x\]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
0
TuringTest
  • TuringTest
incorrect
anonymous
  • anonymous
This integral does not converge

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anonymous
  • anonymous
indeterminate?
lgbasallote
  • lgbasallote
int dx/x = lnx ln( inf) - ln (-inf) inf - inf...l'hospital!!!
TuringTest
  • TuringTest
correct, but why is it not zero? can we make it zero?
anonymous
  • anonymous
ln(-infinity) isn't defined
TuringTest
  • TuringTest
yes, but that's not the main problem
TuringTest
  • TuringTest
that can't be fixed there is a way to fix this integral to make it give zero, as it intuitively should
TuringTest
  • TuringTest
perhaps "fix" is the wrong word but there is something that can be done to get the logical answer of 0
anonymous
  • anonymous
why isn't this zero? the integral is area under the curve of (1/x) from -infty to infty, it must be zero?
TuringTest
  • TuringTest
but we have a singularity at x=0
Zarkon
  • Zarkon
it is not...the integral does not converge for several reasons
TuringTest
  • TuringTest
Yes Zarkon, you taught me about this, I'm trying to see how others take it
Zarkon
  • Zarkon
ic
anonymous
  • anonymous
what about the logical, intuitive way you were talking about? it was a trap?
TuringTest
  • TuringTest
the 'intuition' I would have (before Zarkon enlightened me) is that because 1/x is odd, this is zero but the singularity, amongst other reasons, prevents this trick
anonymous
  • anonymous
okay, this might be a little stupid or a lot stupid to ask but why is \[\int _{\infty} ^{\infty} \frac1x = 0\]
TuringTest
  • TuringTest
\[\int_{-a}^{a}f(x)dx=0\]if f(x) is odd, so one may come to the conclusion that\[\int_{-\infty}^{\infty}\frac{dx}x=0\]as I did in an earlier problem
anonymous
  • anonymous
I think this must be the case if f(x) is even, maybe \[\int_{-a}^{a} f'(x) dx = [f(x)]_{-a}^{a} = f(a) - f(-a)= 0\]
anonymous
  • anonymous
i shouldn't have answered zero as the answer, something is wrong with my head... now when i think zero doesn't even make sense
TuringTest
  • TuringTest
\[\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\]if f(x) is even
TuringTest
  • TuringTest
and yeah, I realized that zero makes no sense too after a couple minutes as well hence I find this whole dilemma interesting
anonymous
  • anonymous
oh yeah, thanks...
TuringTest
  • TuringTest
do you want me to tell you how physicists get away with saying that this integral is 0 ? or do you want to investigate it yourself?
anonymous
  • anonymous
i don't have a clue how physicists do it, it'd be better you tell me
TuringTest
  • TuringTest
as Zarkon said, the integral really doesn't converge (I'm going to say just because of the singularity at x=0) so there is a trick to avoid this called the Cauchy principle value (CPV) that sort of circumvents the point x=0 evenly in both directions from x=0 i.e. they split the integral and make what would be the middle term x=0 into x=-a and x=a respectively http://en.wikipedia.org/wiki/Cauchy_principal_value
TuringTest
  • TuringTest
I think this is a really good thing to know about improper integrals, and I just learned about it, so I wanted to bring it up here :)
TuringTest
  • TuringTest
physicists sometimes use the CPV without stating it, so that is why I mentioned them
anonymous
  • anonymous
thanks, really nice of you to do so... i think some threads on openstudy must be made resource threads or wiki threads maybe (like this one)
TuringTest
  • TuringTest
lol, well it's got the wiki link on it! but thanks, fun discussion :)

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