Results 1 to 5 of 5

Math Help - Is 1/(x^2 - 1) Lebesgue integrable on (0,1)?

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    103

    Is 1/(x^2 - 1) Lebesgue integrable on (0,1)?

    Is 1/(x^2 - 1) Lebesgue integrable over (0,1)? What about over (1, \infty)? What kind of comparisons should I be trying?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    I thought that everything that was Riemann integrable was also Lebesgue integrable.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2009
    Posts
    103
    OK...so why is it Reimann integrable?Thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Oh crap...sorry, I misread the question. I thought it said \frac{1}{x^2+1}.

    \frac{1}{x^2-1} is definitely not Riemann integrable on (0,1). I don't know enough about Lebesgue integration to solve this problem. Sorry for wasting your time.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    The integral of \frac1{x^2-1} diverges at x=1. That is to say, it diverges as an improper Riemann integral. But that implies that it does not exist as a Lebesgue integral, on either of the intervals (0,1) or (1,∞).

    To prove that formally, on the interval (1,∞) you could define a sequence of functions f_n(x) = \begin{cases}\frac n{n+2}&(1<x\leqslant1+\frac1n),\\ \frac1{x^2-1}&(x>1+\frac1n).\end{cases}
    Then f_n(x) increases to \frac1{x^2-1} as n\to\infty, but \int_{(1,\infty)}f_n\to\infty. It follows from the monotone convergence theorem that \frac1{x^2-1} is not (Lebesgue) integrable on (1,∞). A similar argument shows that it is not integrable on (0,1).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Product of Lebesgue integrable functions...
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 17th 2011, 12:38 PM
  2. Is a bounded integrable function square integrable?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 28th 2010, 07:26 PM
  3. Is this function (Lebesgue) Integrable?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 28th 2010, 02:50 AM
  4. [SOLVED] f integrable implies f^2 integrable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: June 8th 2009, 11:53 PM
  5. Are the following Lebesgue integrable?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 11th 2009, 03:07 AM

Search Tags


/mathhelpforum @mathhelpforum