Results 1 to 3 of 3

Math Help - Integral

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    Integral

    Hi,

    How to calculate the following integral \int_{e^{3}}^{e^{4}}\frac{ln(1+\sqrt{x})}{x+\sqrt{  x}} ???
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Apr 2010
    Posts
    57
    Start with integration by parts.

    \int{u*dv}=uv-\int{v*du}

    u=ln(1+\sqrt{x})
    du=\frac{1}{2\sqrt{x}*(\sqrt{x}+1)}
    v=2ln(1+\sqrt{x})
    dv=\frac{1}{x+\sqrt{x}}

    then

    uv=(ln(1+\sqrt{x}))*(2ln(1+\sqrt{x}))=2ln(1+\sqrt{  x})^2

    and

    \int{v*du}=\int{\frac{ln(1+\sqrt{x})}{\sqrt{x}(1+\  sqrt{x}}}

    Do it again and you'll get

    \int{\frac{ln(1+\sqrt{x})}{\sqrt{x}(1+\sqrt{x}}}=(  integration by parts)=ln(1+\sqrt{x})^2)

    and so

    (\int{u*dv}=uv-\int{v*du})=((2ln(1+\sqrt{x})^2)-ln(1+\sqrt{x})^2)))=ln(1+\sqrt{x})^2

    And of course, now that you have done the integral you can enter the parameters and evaluate it.
    Last edited by shenanigans87; April 30th 2010 at 08:17 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jan 2009
    Posts
    715
     \int \frac{\ln(1+\sqrt{x})}{x+\sqrt{x}}~dx

     = \int \frac{1}{\sqrt{x}} \frac{\ln(1+\sqrt{x})}{1+\sqrt{x}}~dx

     = 2 \int  \frac{\ln(1+\sqrt{x})}{1+\sqrt{x}}~d(\sqrt{x})

     = 2\int  \ln(1+\sqrt{x}) ~d(\ln(1+\sqrt{x}))

    sub. u = \ln(1+\sqrt{x})

    The integral

     = 2 \int udu = u^2 + C =  \ln^2(1+\sqrt{x}) + C
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: August 31st 2010, 07:38 AM
  2. Replies: 1
    Last Post: June 2nd 2010, 02:25 AM
  3. Replies: 0
    Last Post: May 9th 2010, 01:52 PM
  4. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 15th 2009, 01:28 PM
  5. Replies: 0
    Last Post: September 10th 2008, 07:53 PM

Search Tags


/mathhelpforum @mathhelpforum