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Integrate x^2 /square root (x^2 +1)

  1. Sep 26, 2009 #1
    Hi,
    the question is integrate x^2 /square root (x^2 +1).

    I tried to do it but i was stucked in the very last step.at the end,I got x/2 * cosh(arcsinh x) + 1/2 * arcsinh x - arcsin (square root (x^2 + 1)) +C. I dunnoe how can I simplify cosh(arcsinh x) into simplier form in term of x. I hope what I wrote is not too confusing.Thanks!
     
  2. jcsd
  3. Sep 26, 2009 #2

    Dick

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    Re: integration

    I haven't checked your integration but arcsinh(z)=ln(z+sqrt(1+z^2)). You can put that into cosh(y)=(e^y+e^(-y))/2 and get an algebraic form for cosh(arcsinh(z)).
     
  4. Sep 26, 2009 #3

    tiny-tim

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    Hi yanjt! :smile:

    (have a square-root: √ and try using the X2 tag just above the Reply box :wink:)

    i] cosh(arcsinh x) is cosh of the "angle" whose sinh is x … so just use cosh2 - sinh2 = 1. :wink:

    ii] why is your result so complicated? just use a substitution. :smile:
     
  5. Sep 26, 2009 #4
    Re: integration

    Thanks, dick n tiny-tim!:smile:

    I did it in this way:

    x2/√(x2+1) = √(x2+1) - 1/√(x2+1)

    Integrate 1/√(x2+1) = arcsinh x
    For √(x2+1),i subsitute x=sinh t.
    So, integrate √(x2+1 = cosh2t dt
    = 1/4 sinh 2t + t/2
    =1/4 (2 sinh t cosh t) + arcsinh x/2
    =x/2 (cosh(arcsinh x)) + arcsinh x/2
     
  6. Sep 26, 2009 #5

    tiny-tim

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    Hi yanjt! :smile:

    Wouldn't it have been easier to substitute x = sinht at the very start? :wink:

    (and you can't leave cosh(arcsinh x) as it is … you must convert that into something more "algebraic" :smile:)
     
  7. Sep 26, 2009 #6
    Re: integration

    ooohhhh yeah!y i didnt think of tat?:confused: tat's a faster n easier way to do it!thanks!!:smile: but is integrate 1/cosh x = 1/sinh x?can i jus integrate like tat?

    i know tat i cant jus leave cosh(arcsinh x) as my final answer.tat's y i m stucked in this step.:grumpy: but i will try to change it into somethg more "algebraic" using the methods tat dick n u mentioned in the first 2 posts.
     
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