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Integration (maybe by parts)

  1. Oct 20, 2012 #1
    1. The problem statement, all variables and given/known data
    $$\int \frac{x}{(1-x^2)} dx$$


    2. Relevant equations

    Integration by parts, by substitution, etc.

    3. The attempt at a solution

    I just can't remember how to begin this integration. I tried doing integration by parts, where
    $$a(x) = x$$
    $$a'(x) = 1$$
    $$b(x) = $$
    $$b'(x) = \frac{1}{(1-x^2)}$$
    $$\int \frac{x}{(1-x^2)}dx = xb(x) - \int \frac{1}{(1-x^2)} dx$$

    But couldn't work out what form b(x) should take.
    Thanks in advance for any help
     
  2. jcsd
  3. Oct 20, 2012 #2
    If you have seen enough of these integrals, they can essentially done "by observation"

    If not, substitution is a good method to use. Substitute [itex]y = x^{2}[/itex]. Then [itex]dy = 2x dx[/itex]
    Then the integral becomes:
    [tex]\int \frac{1}{2}\frac{1}{1-y}dy[/tex]

    Does that now look more doable?
     
    Last edited: Oct 20, 2012
  4. Oct 20, 2012 #3

    HallsofIvy

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    Why not let u= 1- x^2 itself? Then du= -2xdx so that xdx= -(1/2)du do that
    [itex]-\frac{1}{2}\int \frac{du}{u}[/itex]
     
  5. Oct 20, 2012 #4
    Ah yes, thanks...that makes sense. I wasn't getting where the sqrt cam from in the first response.
    Thanks to both of you
     
  6. Oct 20, 2012 #5
    Oops, typo, my bad.
     
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