Results 1 to 3 of 3

Math Help - Improper Integration

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Improper Integration

    \displaystyle\int_{-\infty}^{\infty}\frac{x}{\pi(1+x^2)} \ dx=\left[\frac{\ln|1+x^2|}{2\pi}\right]_{-\infty}^{\infty}

    How do I evaluate this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    See this.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by dwsmith View Post
    \displaystyle\int_{-\infty}^{\infty}\frac{x}{\pi(1+x^2)} \ dx=\left[\frac{\ln|1+x^2|}{2\pi}\right]_{-\infty}^{\infty}

    How do I evaluate this?
    The question is a little controversial, so that it is better, if You want the 'Cauchy principal value' of integral, to specify that as follows...

    \displaystyle \text{PV} \int_{-\infty}^{+\infty} \frac{x}{\pi\ (1+x^{2})}\ dx = \lim_{\xi \rightarrow \infty} \int_{-\xi}^{\+\xi} \frac{x}{\pi\ (1+x^{2})}\ dx =0

    Without the prefix 'PV' the integral...

    \displaystyle \int_{-\infty}^{+\infty} \frac{x}{\pi\ (1+x^{2})}\ dx

    ... has to be considered divergent...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Improper Integration
    Posted in the Calculus Forum
    Replies: 6
    Last Post: April 17th 2010, 09:17 PM
  2. improper integration by parts
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 9th 2009, 05:52 AM
  3. Improper Integration
    Posted in the Calculus Forum
    Replies: 4
    Last Post: December 3rd 2009, 10:45 PM
  4. Improper Integration
    Posted in the Calculus Forum
    Replies: 9
    Last Post: June 22nd 2008, 12:33 PM
  5. improper integration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 15th 2007, 06:52 PM

/mathhelpforum @mathhelpforum