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Homework Help: Intergration problem

  1. Feb 8, 2009 #1
    1. The problem statement, all variables and given/known data
    i'm taking a defitinite integral from sqrt2 to 2 of the function 1/x^3*sqrt(x^2-1)dx.

    2. Relevant equations



    3. The attempt at a solution
    I seperated it into 1/x^3 and 1/sqrt(x^2-1). I have the second part using trig sub. as being sec theta dtheta, before integrating it. I believe i did this part correctly.

    What I can't remember is that I make 1/x^3 to x^-3 and then integrate it that way with the final being -1/2(1/x^2)??
     
  2. jcsd
  3. Feb 8, 2009 #2

    Dick

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    I know the forums are slow, but you can stop posting this now. The substitution you want is u^2=x^2+1. Work it out. You can turn it into a rational function in u. Then use partial fractions.
     
  4. Feb 8, 2009 #3
    for some reason this forum isn't working, and i didn't mean to repost several times... I hit refresh a couple times and that's what happened. I apologize!
     
  5. Feb 8, 2009 #4

    Dick

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    S'ok. Can you remove the other ones or mark as duplicates?
     
  6. Feb 8, 2009 #5
    Ok, was i correct in splitting it into two integrals?

    I'm having alot of difficulty on this one. I got an answer, but it's really long and seems wrong.

    I found I'm supposed to get but need to be able to work through this problem.


    How i got mine was for the second part used x = sec y and dx= sec y tan y dy.
    after working the substitution i came up with the integral of sec y dy from sqrt(2) to 2. the integration of sec is ln abs(sec y + tan y). I then substituted the inverse sec (x/a) in for y based on my original substitution.

    that is then multiplied by the integral of 1/x^3 which is -1/(2x^2). Was this the correct way in doing it?
     
  7. Feb 8, 2009 #6

    Dick

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    You can't 'split it into two integrals'. There's no rule that says integral(f*g)=integral(f)*integral(g). That's just plain wrong. You can probably handle it with a trig substitution as well, but I would suggest using my original suggestion of u^2=x^2+1.
     
  8. Feb 8, 2009 #7
    ok. do you mean to say substitute u^2=x^2 -1 to get rid of the sqrt then?

    i just don't understand how that helps me. I end up with variables x and u in the denominator then.
     
  9. Feb 8, 2009 #8

    Dick

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    Yes, I meant to say u^2=x^2-1. Sorry. It still works. The stray x will cancel out. x^2=u^2+1.
     
    Last edited: Feb 9, 2009
  10. Feb 9, 2009 #9
    what would x^3 be in terms of u? I'm gonna have to head to a tutor and have someone walk me through this one.
     
  11. Feb 9, 2009 #10
    Here is a hint: don't express x3 in terms of u. At first, express only the square root term and dx in terms of u and du. This will give you an easier term than x^3 to express in terms of u.
     
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