Results 1 to 3 of 3

Thread: Techniques of integration (4)

  1. #1
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Techniques of integration (4)

    Evaluate $\displaystyle I=\int \frac{x(2x-1)}{(x^2-x+1)^2 + 1} \ dx.$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
    I think you wanna test whether we are brave enough to expand and factorize the denominator

    $\displaystyle (x^2 - x + 1)^2 + 1 $
    $\displaystyle = x^4 -2x^3 + 3x^2 - 2x + 1 + 1 $
    $\displaystyle = x^4 -2x^3 + 3x^2 - 2x + 2 $
    $\displaystyle = ( x^4 + 3x^2 + 2 ) - 2x(x^2 + 1)$
    $\displaystyle = ( x^2 + 1 )(x^2 + 2) - 2x( x^2 + 1)$
    $\displaystyle = (x^2 + 1)(x^2 - 2x + 2)$

    and

    $\displaystyle \frac{x(2x-1)}{(x^2 - x + 1)^2 + 1}$

    $\displaystyle = \frac{-x}{x^2 + 1} + \frac{x}{x^2 - 2x + 2}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by simplependulum View Post
    I think you wanna test whether we are brave enough to expand and factorize the denominator

    $\displaystyle (x^2 - x + 1)^2 + 1 $
    $\displaystyle = x^4 -2x^3 + 3x^2 - 2x + 1 + 1 $
    $\displaystyle = x^4 -2x^3 + 3x^2 - 2x + 2 $
    $\displaystyle = ( x^4 + 3x^2 + 2 ) - 2x(x^2 + 1)$
    $\displaystyle = ( x^2 + 1 )(x^2 + 2) - 2x( x^2 + 1)$
    $\displaystyle = (x^2 + 1)(x^2 - 2x + 2)$

    and

    $\displaystyle \frac{x(2x-1)}{(x^2 - x + 1)^2 + 1}$

    $\displaystyle = \frac{-x}{x^2 + 1} + \frac{x}{x^2 - 2x + 2}$
    and you were brave enough! well done!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Techniques of integration (6)
    Posted in the Math Challenge Problems Forum
    Replies: 4
    Last Post: May 22nd 2009, 05:54 PM
  2. Techniques of integration (5)
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: May 20th 2009, 06:09 AM
  3. Techniques of integration (3)
    Posted in the Math Challenge Problems Forum
    Replies: 9
    Last Post: May 17th 2009, 03:55 PM
  4. Techniques of integration (2)
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: May 2nd 2009, 06:09 AM
  5. Techniques of integration (1)
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: Apr 24th 2009, 06:50 PM

Search Tags


/mathhelpforum @mathhelpforum