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Throwing everything I can at this one

  1. Sep 23, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\int \frac{(arcsinx)^{2}}{\sqrt{1-x^{2}}} dx[/itex]

    2. Relevant equations

    3. The attempt at a solution

    First I attempted by parts, that got nowhere. Then, I attempted a u substitution for u = arcsin x, and when that did nothing I attempted one for u = arcsinx^2.

    I have no idea what to do with this one. How should I approach the problem?
    Last edited: Sep 23, 2011
  2. jcsd
  3. Sep 23, 2011 #2


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    Is the problem [itex]\int \frac{(arctanx)^{2}}{\sqrt{1-x^{2}}} dx[/itex] or [itex]\int \frac{(arcsinx)^{2}}{\sqrt{1-x^{2}}} dx[/itex]?
  4. Sep 23, 2011 #3
    arcsin, definitely. Sorry about that.
  5. Sep 23, 2011 #4


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    Your original intuition looks fine. Set u = arcsinx. du = ...?
  6. Sep 23, 2011 #5
    My original intuition of integration by parts?? I didn't think that was the preferred method because I'd have to integrate (1-x^2)^(-1/2) which would just complicate it
  7. Sep 23, 2011 #6
    Ok wow, this is an extremely simple substitution problem, provided you actually remember the derivative has a square root in the denominator.

    Try doing it thinking it's like the arctan derivative and you will see my frustration.
  8. Sep 23, 2011 #7


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    If it was, I have no idea how to solve that. You probably won't make that mistake again.
  9. Sep 23, 2011 #8
    Numerically. :wink:
  10. Sep 23, 2011 #9


    Staff: Mentor

    Not really a solution for an indefinite integral...
  11. Sep 23, 2011 #10
    Touche, I guess...didn't look that closely and missed the lack of limits.
  12. Sep 23, 2011 #11
    Integration is getting really difficult for me with all the new techniques I had to learn in like 2 days.

    How do you guys decide which method to try first?
  13. Sep 23, 2011 #12


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    Usually depending on how difficult it looks, you should try to do substitution before you attempt integration by parts.

    For example, in your question, you can see a relation between arcsin(x) and 1/sqrt(1-x^2), substitution might work here.
  14. Sep 23, 2011 #13
    I have another question.

    One of my favorite integrals to solve is actually integral of the arctangent. I like it because it's kind of neat, integrate by parts once, then u-substitution once, it looks good on paper.

    However, you need to know it's derivative when you call it u for integration by parts.

    I just have it memorized. I don't really intuitively understand how one arrives at the derivative of arctangent. Is it just something that is observable or can it actually be shown?
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