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Math Help - Intergrate sin(x^2) or cos(x^2)?

  1. #1
    Senior Member TriKri's Avatar
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    Intergrate sin(x^2) or cos(x^2)?

    Is it possible to intergrate sin(x^2) or cos(x^2)?
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    Quote Originally Posted by TriKri View Post
    Is it possible to intergrate sin(x^2) or cos(x^2)?
    \int_0^{\infty}\sin (x^2) dx = \int_0^{\infty} \cos (x^2) dx = \sqrt{\frac{\pi}{8}}.
    That is the best I can do.
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    Senior Member TriKri's Avatar
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    Hm, it's not infinite then? It's not like sin(x^2)\rightarrow 0 when x\rightarrow \infty, just a thought though, it's probably true if you say so.

    By the way, I had more finite integrals in mind. There's not a primitive function or something?
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    Quote Originally Posted by TriKri View Post
    Hm, it's not infinite then? It's not like sin(x^2)\rightarrow 0 when x\rightarrow \infty, just a thought.

    By the way, I had more finite integrals in mind. Like a primitive function or something.
    So? There is no such theorem.
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    Senior Member TriKri's Avatar
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    Oh, I didn't realize it staibilizes because sinus and cosinus hovers between -1 and 1, the average will \rightarrow 0.

    But there can't be any primitive function, right? Cause I else I would have found a way to integrate the Gaussian function, and I guess it has been proven that there's no way to do it.
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    Quote Originally Posted by TriKri View Post
    Cause I else I would have found a way to integrate the Gaussian function, and I guess it has been proven that there's no way to do it.
    No there is not way to integrate the Gaussian integral. This is a famous result. However the integral \int_{-\infty}^{\infty}e^{-x^2} dx = \sqrt{\pi} is a well-known and important result. The beauty of infinite integrals is that you can give a closed form answer though you cannot find the primitive.
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    Senior Member TriKri's Avatar
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    Then I will stop try to integrate it and instead realise it's beauty.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by TriKri View Post
    Is it possible to intergrate sin(x^2) or cos(x^2)?
    Here is what WebMathematics says:
    Attached Thumbnails Attached Thumbnails Intergrate sin(x^2) or cos(x^2)?-gash.jpg  
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    Quote Originally Posted by CaptainBlack View Post
    Here is what WebMathematics says:
    There are many functions which do not have a closed form integral in terms
    of elementary functions (in some sense these are more common that those
    that do, but we don't tell you that, we only teach what can be done in
    terms of elementary functions). The usual technique to deal with an
    important class of integrals like this is to introduce a new class of special
    functions in terms of which the integral can be written in cloded form.

    An example which you may have met is the integral of the Gaussian density,
    here we introduce the error function which allows us to write the integral in
    closed form.

    You might find this article of interest.

    RonL
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