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Math Help - How to integrate sqrt(arctanx)?

  1. #1
    Junior Member NowIsForever's Avatar
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    How to integrate sqrt(arctanx)?

    This question is posed here: How to integrate sqrt(arctanx)? - Yahoo! Answers

    I think the presumption of the only answer by Robert V is correct and that the OP got the order of operation mixed up.

    However, maybe someone here can provide a closed form answer to the original question?

    (If not, then thanks for your efforts.)
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  2. #2
    Senior Member x3bnm's Avatar
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    Re: How to integrate sqrt(arctanx)?

    Quote Originally Posted by NowIsForever View Post
    This question is posed here: How to integrate sqrt(arctanx)? - Yahoo! Answers

    I think the presumption of the only answer by Robert V is correct and that the OP got the order of operation mixed up.

    However, maybe someone here can provide a closed form answer to the original question?

    (If not, then thanks for your efforts.)
    For:
    \int \tan^{-1}(\sqrt{x})\,\,dx

    Let s = \sqrt{x} \therefore ds = \frac{1}{2\sqrt{x}}\,\,dx = \frac{1}{2s}\,\,dx

    So dx = 2s\,\,ds

    \begin{align*}\int \tan^{-1}{(\sqrt{x})}\,\,dx = \int 2s\tan^{-1}{(s)}\,\,ds....\text{[from above]}\end{align*}



    \text{Let } u = \tan^{-1}{(s)} \therefore du = \frac{1}{1+s^2}\,\, ds

    \text{And } dv = 2s\,\, ds \therefore v = s^2


    Now using integration by parts:

    \begin{align*}&\int 2s\tan^{-1}{(s)}\,\,ds \\=&  s^{2}\tan^{-1}{(s)} - \int \frac{s^2}{1+s^2}\,\,ds\\=& s^{2}\tan^{-1}{(s)} - \int \left(1 - \frac{1}{1+s^2}\right)\,\, ds\\=& s^{2}\tan^{-1}{(s)} -  \int \,\,ds +  \int \frac{1}{1+s^2}\,\,ds\\=& s^{2}\tan^{-1}{(s)} - s + \tan^{-1}(s) +C \\ =& x\tan^{-1}{(\sqrt{x})} - \sqrt{x} + \tan^{-1}(\sqrt{x}) + C.....\text{[because }s = \sqrt{x}\text{]}\end{align*}

    You can check if my answer is correct here:
    int arctan((x)^(1/2)) - Wolfram|Alpha

    Hope this helps.
    Last edited by x3bnm; August 21st 2013 at 12:43 PM.
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  3. #3
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    Re: How to integrate sqrt(arctanx)?

    [QUOTE

    Hope this helps.[/QUOTE]

    It doesn't, read the question.
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  4. #4
    Junior Member NowIsForever's Avatar
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    Re: How to integrate sqrt(arctanx)?

    Thanks, x3bnm, but you have answered the reverse question just as Robert V did in Yahoo! Answers. The original (albeit, possibly mistaken) question was int sqrt(arctan(x))?

    wolframalpha.com couldn't do it either, though perhaps the Pro version could.

    I suspect there is no closed form solution, although proving that could be incredibly difficult.
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