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Easy integral from exam

  1. Sep 1, 2011 #1
    1. The problem statement, all variables and given/known data

    From an exam in calc 2 we are reviewing simple integrals. This one was annoying because it actually contained algebra.. regardless.

    [itex]\int(\frac{1 - y^{2}}{y^{2}})^{2} dy[/itex]

    2. Relevant equations

    3. The attempt at a solution

    First I broke it into two fractions, and turned the second into 1 as it is y squared over y squared.

    [itex]\int(\frac{1}{y^{2}} - 1)^{2} dy[/itex]

    Then squared the polynomial of sorts.. to get

    [itex]\int y^{-4} - y^{-2} - y^{-2} + 1 dy[/itex]

    Leading me to a final answer of

    [itex]- \frac{y^{-3}}{3} + 2y^{-1} + y + C[/itex]

    Look okay? A bit rusty in algebra..
  2. jcsd
  3. Sep 1, 2011 #2
    Looks fine to me!
  4. Sep 1, 2011 #3


    Staff: Mentor

    There's a mistake in your 2nd term. The coefficient of the y-1 term should be 1, not 2.

    Also, a slightly different approach is to square the numerator and denominator of your fraction instead of doing the division first. This leads to the same result, though, so can't really be considered a better approach.
  5. Sep 1, 2011 #4
    How so? After integration of -y^-2 I get +y^-1, and there are two instances of -y^-2. All I did was add them together for 2y^-1.
  6. Sep 1, 2011 #5
    His integral shows that he is adding y^-2 to y^-2; he just fails to simplify before he integrates. I think that's where his 2 comes from.
  7. Sep 1, 2011 #6


    Staff: Mentor

    Sorry, I totally missed that there was another y-1 term. My mistake...
  8. Sep 1, 2011 #7
    No problem, thanks guys!
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