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Math Help - Trouble with seemingly trivial integral

  1. #1
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    Trouble with seemingly trivial integral

    Hello everyone,

    I have been having a lot of trouble from a seemingly simple integral:
    <br />
\int arctan (arctan x)dx<br />
.

    I say that it's trivial because arctan (arctan x) is a function that can be plotted and does not go to infinity anywhere. In fact, it looks more or less like arctan, which does have an integral.

    Trying to plug it into mathematica yields no results, so I was wondering if there was an integral, and if so, how it should be approached.

    I tried integration by parts, substitution, and integration by parts again, but all I ended up with was \int sec^2 (u) * arctan (u) du, where x=tan u.

    Assistance would be very greatly appreciated!

    P.S. Attached is work done and graphs of some important functions as mentioned above, but really getting nowhere...

    Trouble with seemingly trivial integral-problem.pdf

    Trouble with seemingly trivial integral-graphs.pdf
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  2. #2
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    Quote Originally Posted by progressive View Post
    Trouble with seemingly trivial integral
    It isn't trivial. I doubt it could be expressed in a closed form. (I distinctly remember it being stated that no
    known elementary anti-derivative of sin(sin(x)) exists and this isn't much different). If you want to evaluate
    it over a given domain, then there are methods for finding that without finding the anti-derivative.
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  3. #3
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    TheCoffeeMachine,

    Thanks for your help. I want to understand why sin(sin(x)) or arcsin(arcsin(x)) cant be integrated indefinitely a little better. Do you by any chance remember where you heard/read that sin(sin(x)) does not exist?

    Also, it may be my particular understanding of an integral... which functions don't go to infinity yet don't have an integral? That is, why is calculating the indefinite integral of a "clean" function (as seen through graphing) not trivial? Our teacher hasn't covered them yet.

    Thanks again.
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  4. #4
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    Quote Originally Posted by progressive View Post
    Do you by any chance remember where you heard/read that sin(sin(x)) does not exist?
    In this book, here.

    Also, it may be my particular understanding of an integral... which functions don't go to infinity yet don't have an integral? That is, why is calculating the indefinite integral of a "clean" function (as seen through graphing) not trivial? Our teacher hasn't covered them yet.
    I'm not sure I entirely understand what you mean here, but a f(x) = e^{-x^2} is a continuous function that
    you can easily graph yet it has no elementary anti-derivative. The same goes for x^x and \frac{\sin{x}}{x}. I think the
    simplest way to understand the reason is that there isn't an elementary function whose derivative gives
    any of these functions. It has very little to do with continuity or the nicety of their graphs, I think.
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