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Math Help - Integral help

  1. #1
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    Integral problem with arctan

    I need a help integrating:
    <br />
\int Arctan^2xdx<br />
    Any ideas?
    Last edited by Toxic; March 7th 2008 at 06:03 AM.
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  2. #2
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    Krizalid's Avatar
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    I'm not gettin' anything nice.

    Could it be \arctan x^2? At least that is not so ugly.
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  3. #3
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    Nah \int Arctan x^2dx is easy :P. I need help with the other one and indeed it is an ugly one.
    The best i got so far is this:

    with partial intergration:

    u = (arctan x)^2, du = \frac{2(arctan x)}{(1 + x^2)}
    dv = dx,v = x

    and im getting:

    \int arctan^2 x dx= x*(arctan^2 x) - \int\frac{2x*(arctan x)}{1 + x^2} dx

    Now im integrating \int\frac{2x*(arctan x)}{1 + x^2}dx with partial:

    u = arctan x,du = \frac{1}{1 + x^2}
    dv = \frac{2x}{1 + x^2},v = ln(1 + x^2)

    and im getting:

    \int\frac{2x*(arctan x)}{1 + x^2} dx<br />
= ln(1 + x^2)*(arctan x) - \int\frac{ln(1 + x^2)}{1 + x^2} dx


    and im stuck at this last integral
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  4. #4
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    Krizalid's Avatar
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    The last integral hasn't elementary primitive. It's useless to keep workin' out this.
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  5. #5
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    But is there another way to do it? without getting the logarithmic integral?
    Maybe some different partial integration or something?
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  6. #6
    Math Engineering Student
    Krizalid's Avatar
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    I don't think so. I put your integral to check it with a software and its answer involves complex numbers.

    There're lots of definite integrals which don't have elementary primitives, but workin' with integration limits one can find a suitable answer. In this case, I tried with some boundary limits but nothing worked out.
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