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Difficult Integral

  1. Jan 24, 2013 #1
    1. The problem statement, all variables and given/known data
    [tex]\int (arcsin x)^{2}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    [tex]u=arcsin x[/tex] [tex]du=1/\sqrt{1-x^{2}}dx[/tex]
    [tex]v=?[/tex] [tex]dv=arcsin x dx[/tex]
     
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  3. Jan 24, 2013 #2

    Dick

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    Re: Integral

    To integrate arcsin(x)dx do parts again. u=arcsin(x) dv=dx.
     
  4. Jan 24, 2013 #3
    Re: Integral

    Ok then I get:
    [tex]xarcsinx-\int x/(\sqrt{1-x^{2}})[/tex]
     
  5. Jan 24, 2013 #4

    Dick

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    Re: Integral

    The second term can be integrated with a simple substitution.
     
  6. Jan 24, 2013 #5
    Re: Integral

    Ok after all that:
    [tex]\int arcsinx=xarcsinx+\sqrt{1-x^{2}}[/tex]
    Then putting in for v, I end up with:
    [tex](arcsinx(arcsinx+\sqrt{1-x^{2}})-\int \frac{(xarcsinx+\sqrt{1-x^{2}})}{(\sqrt{1-x^{2}})}[/tex]

    Do I do another u-substition for the integral, u=arcsinx? But I don't know what to do with the x infront of it.
     
  7. Jan 24, 2013 #6

    Zondrina

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    Re: Integral

    No no!

    Let u = 1 - x2 so that du = -2x dx which means -du/2 = x dx.

    What does that second integral x / sqrt(1-x2) become now?
     
  8. Jan 24, 2013 #7
    Re: Integral

    That doesn't work or I'm not understanding
     
  9. Jan 24, 2013 #8

    Zondrina

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    Re: Integral

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

    [itex] u = 1-x^2 → - \frac{1}{2} du = xdx[/itex]

    [itex]\int \frac{x}{\sqrt{1-x^2}} dx = - \frac{1}{2} \int \frac{1}{\sqrt{u}} du[/itex]
     
  10. Jan 24, 2013 #9

    Dick

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    Re: Integral

    Split the integral into two parts. Looks like you need another round of integration by parts on the first piece.
     
  11. Jan 24, 2013 #10
    Re: Integral

    The integral I'm trying to solve is:
    [tex]\int \frac{xarcsinx+\sqrt{1-x^2}}{\sqrt{1-x^{2}}}dx[/tex]
     
  12. Jan 24, 2013 #11

    Zondrina

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    Re: Integral

    Oh my apologies, I thought :

    Was what you were having trouble with.
     
  13. Jan 24, 2013 #12
    Re: Integral

    Lol all good, I was like am I being stupid or what?
     
  14. Jan 24, 2013 #13
    Re: Integral

    This just never ends omg.
     
  15. Jan 24, 2013 #14

    Dick

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    Re: Integral

    You are almost there. It's downhill from here.
     
  16. Jan 24, 2013 #15
    Re: Integral

    I don't know the derivative of xarcsinx
     
  17. Jan 24, 2013 #16

    Dick

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    Re: Integral

    u=arcsin(x) dv=xdx/sqrt(1-x^2). Integrating that should look familiar.
     
  18. Jan 24, 2013 #17
    Re: Integral

    Yeah I got it,
    My answer comes out:
    [tex]arcsinx(arcsinx+\sqrt{1-x^{2}})-(\frac{x(arcsinx)^{2}}{2}+x)[/tex]
     
  19. Jan 24, 2013 #18

    Dick

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    Re: Integral

    Not right I'm afraid. I think you might have goofed up the last integration by parts. Check it.
     
  20. Jan 24, 2013 #19
    Re: Integral

    This is ridiculous so much work just to get the wrong answer lol.

    I got what I did wrong, it was the last integration by parts.
     
  21. Jan 24, 2013 #20

    Dick

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    Re: Integral

    Now that you know the steps you just have to make sure everything is right. You're close.
     
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