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Math Help - Messy Integral!

  1. #1
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    Messy Integral!

    Hey all.

    I was trying to figure out a way to evaluate the following integral:

    INT ( ln(lnx) + 1/[(lnx)^2] ) dx

    The answer to the above is:

    x ln(lnx) - [ x / (lnx) ] + c

    My question is, how to get to it?

    Any help/hints/suggestions will be greatly appreciated.

    Thanks,

    Shahz.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    You can try a substitution of

    e^u = ln(x)

    I think that will make things simpler.
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  3. #3
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    Quote Originally Posted by Unknown008 View Post
    You can try a substitution of

    e^u = ln(x)

    I think that will make things simpler.
    Thanks for this but I tried it and I end up with two messy looking integrals in the end involving u and e^u.

    Can't we just use integration by parts?

    Help!!
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  4. #4
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    Yes you can - integrate ln(ln x) by parts twice, integrating 1 each time (and differentiating the integrand each time)... see what happens. Pic in a mo.

    Edit: sorry for the delay. Just in case a picture helps...



    ... where



    ... is lazy integration by parts, with...



    ... the product rule. Straight lines differentiate downwards (integrate up) with respect to x.

    Spoiler:

    Hence we can write...
    Spoiler:



    You may find it helpful to zoom in on the chain rule, inside the product rule...



    Spoiler:

    ... where



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).



    _______________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; July 4th 2010 at 02:56 PM.
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  5. #5
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    Quote Originally Posted by tom@ballooncalculus View Post
    Yes you can - integrate ln(ln x) by parts twice, integrating 1 each time (and differentiating the integrand each time)... see what happens. Pic in a mo.

    Edit: sorry for the delay. Just in case a picture helps...



    ... where



    ... is lazy integration by parts, with...



    ... the product rule. Straight lines differentiate downwards (integrate up) with respect to x.

    Spoiler:

    Hence we can write...
    Spoiler:



    You may find it helpful to zoom in on the chain rule, inside the product rule...



    Spoiler:

    ... where



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).



    _______________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Wow!! Where do I start and what do I say. You're simply fantastic Tom. Thank you so much. I kind of figured it out before you posted the pics but they're a visual delight. I loved it when the two integrals involving the messy 1 / (lnx)^2 just cancelled out. However, if we just had the integral of ln(lnx), I don't think it can be done with elementary functions.
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  6. #6
    MHF Contributor Unknown008's Avatar
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    Yes, sorry, I made a mistake. I think that substituting x = e^u is better.

    This done, I get:

    \int e^u ln(u) du + \int \frac{e^u}{u^2} du

    If I didn't do any mistake

    Then, I proceeded using by parts.

    I wanted to edit my post yesterday, but my connection was not working properly

    For the drawing, it's the first time I see this... it might take me some time before understanding it

    Anyway, sorry for my first post, I thought it would work without actually trying
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