Results 1 to 4 of 4

Math Help - Residues

  1. #1
    Member
    Joined
    Feb 2009
    From
    Chennai
    Posts
    148

    Residues

    Evaluate \int\limits_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}}\ dx
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by Chandru1 View Post
    Evaluate \int\limits_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}}\ dx
    Hint:

    Let  a=\frac13 and  x=e^t \implies dx = e^tdt

    Thus  \int_0^\infty \frac{x^a}{x^2+1}\ dx = \int_{-\infty}^\infty \frac{e^{(a+1)t}}{e^{2t}+1}\ dt
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jan 2009
    Posts
    715
    Let  x= t^3

    The integral

     I = 3 \int_0^{\infty} \frac{ t^3 }{1+ t^6} ~dt

    Sub  t = 1 /s

     I = 3 \int_0^{\infty} \frac{s}{1+s^6}~ds

    Sub  s^2 = u

     I = \frac{3}{2} \int_0^{\infty} \frac{du}{1+u^3}

     I = \frac{3}{2} \int_0^{\infty} \frac{ 1+ u - u}{1+u^3}~du

     I = \frac{3}{2} \left[ \int_0^{\infty} \frac{1+u}{(1+u)(1-u+u^2)}~du - \int_0^{\infty} \frac{u}{1+u^3}~du \right]

    But by substituting  u= 1/y in the integral \int_0^{\infty} \frac{du}{1+u^3} , we have

     \int_0^{\infty} \frac{du}{1+u^3} = \int_0^{\infty} \frac{u du}{1+u^3}

    Thus we have

     \int_0^{\infty} \frac{u du}{1+u^3} = 2I/3

     2I = \frac{3}{2} \int_0^{\infty} \frac{du}{1-u+u^2}

     I = \frac{3}{4} \cdot \frac{4\pi}{3\sqrt{3}}

     I = \frac{\pi}{\sqrt{3}}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    This is a much better way than my method! It requires no complex analysis.

    My way does generalize this problem though:

    Given  -1<\Re(a)<1 ,  \int_0^\infty \frac{x^a}{x^2+1}\ dx = \frac12\pi\sec\left(\frac{\pi a}{2}\right)

    p.s. May I ask how you came up with your solution?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. About sum of residues
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: May 18th 2009, 11:54 AM
  2. Set of residues
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 29th 2009, 08:10 AM
  3. residues
    Posted in the Calculus Forum
    Replies: 6
    Last Post: February 11th 2009, 08:44 AM
  4. residues 2
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 10th 2009, 08:19 AM
  5. Residues
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 6th 2008, 09:29 AM

Search Tags


/mathhelpforum @mathhelpforum