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Math Help - Integration problem

  1. #1
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    Integration problem

    Hi, dear all:

    I'm wondering how to integrate (exp(-sin(x)^2))*sin(x) with respect to x? I believe it has something to do with error function, but just can't figure out a right form. Pleas help me! Thank you very much.

    simonwei
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  2. #2
    Super Member Deadstar's Avatar
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    I'll post up where I'm at with this. Perhaps you will see it before I do.

    Let t=\sin(x), dt = \cos(x)dx.

    Integral becomes,

    \int \frac{e^{-t^2}t}{\cos(\arcsin(t))} dt = \int \frac{e^{-t^2}t}{\sqrt{1-t^2}} dt.

    I further simplified this by letting y=t^2, dy = 2tdt.

    Then we get...

    \int  \frac{e^{-t^2}t}{\sqrt{1-t^2}} dt = \frac{1}{2} \int  \frac{e^{-y}}{\sqrt{1-y}} dy.

    However I can't advance past this yet! Perhaps you can.
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  3. #3
    Super Member
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     e^{-\sin^2(x)}\sin(x) = e^{-1 + \cos^2(x) }\sin(x)

     = e^{-1} e^{\cos^2(x) }\sin(x)

    Sub.  \cos(x) = t ~~~ -\sin(x)~dx = dt

    It becomes  -e^{-1 }\int e^{t^2}~dt

    It is not the error function but it is close , seemingly no one can use a finite number of elementary functions to express this integral ...
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  4. #4
    Super Member Deadstar's Avatar
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    Nice solution simplependulum.

    This is what I further thought of at 4am!

    Let y=t^2 - 1 => t^2 = y+1

    2tdt = dy

    Integral becomes...

    \int \frac{e^{-t^2}t}{\sqrt{1-t^2}}dt = \frac{1}{2} \int \frac{e^{-y}e^{-1}}{\sqrt{-y}} <br />

    = \frac{-ie^{-1}}{2} \int e^{-y}y^{-\tfrac{1}{2}} dy

    = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{y}\Big{)}

     = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{t^2 - 1}\Big{)}

     = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{\sin^2(x) - 1}\Big{)}

     = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{1 - \cos^2(x) - 1}\Big{)}

     = \frac{-i e^{-1} \sqrt{\pi}}{2} erf(i\cos(x))


    EDIT: Apparently that minus should not be there (or there should be a minus in front of the i\cos(x)) but I can't figure out what I've done wrong after skimming over it.
    Last edited by Deadstar; June 10th 2010 at 05:30 AM.
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