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Evaluate this integral

  1. Jun 16, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    [itex]\int \left( \dfrac{ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx [/itex]

    3. The attempt at a solution
    Let x^(1/6) = t

    [itex]\int \dfrac{ln(1+t)t^3}{1+t} dt [/itex]
     
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  3. Jun 16, 2013 #2

    Simon Bridge

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    You want to evaluate:
    $$\int \left( \dfrac{\ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx$$

    You attempted it by doing the substitution: ##x^{(1/6)} = t## - which gets you:

    $$\int \dfrac{\ln(1+t)t^3}{1+t} dt$$
    ... and, from there, you get stuck?

    Are you missing a factor of 6 in there?

    Have you tried putting u=1+t ?
     
  4. Jun 17, 2013 #3

    utkarshakash

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    I forgot that 6. By the way it won't make my life easier. I tried substituting z=1+t as well and came up with this.

    [itex]\displaystyle \int \dfrac{lnz(z-1)^3}{z} dz[/itex]

    Also, how do you get your integral sign bigger? Mine renders as small when I use \int. Here I've used \displaystyle to get it bigger.
     
    Last edited: Jun 17, 2013
  5. Jun 17, 2013 #4

    Simon Bridge

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    If you want to know how I get a particular format for something, use the "quote" button at the bottom of my posts - it shows you the markup.

    Your int signs render small because you are using in-line math style - the "itex" or double-hash tags. To get things to work better, use the displaymath style - the "tex" or double-dollar tags.

    Similarly, you can get standard functions to render properly by putting a backslash in front of them ... like \ln for the natural logarithm.

    After the substitution you should have something of form ##\int f(z)\ln|z|\; dz## ...

    Arm yourself with a bunch of tables to help your strategy:
    http://en.wikipedia.org/wiki/List_of_integrals_of_logarithmic_functions
    http://en.wikipedia.org/wiki/List_of_logarithmic_identities
     
  6. Jun 17, 2013 #5
    Or you can use partial integration to massage the logarithm out (although things will get slightly messy)
     
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