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Thread: integral

  1. December 19th 2011, 01:09 PM #1
    Random Variable
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    integral

    \int_{0}^{\infty} \frac{e^{\cos x} \sin (\sin x)}{x} \ dx = \frac{\pi}{2} (e-1)


    It's tempting to say that \int_{0}^{\infty} \frac{e^{\cos x} \sin (\sin x)}{x} \ dx = \text{Im} \int_{0}^{\infty} \frac{e^{e^{ix}}}{x} \ dx .


    But \int_{0}^{\infty} \frac{e^{e^{ix}}}{x} \ dx diverges.


    EDIT: A principal value approach works.
    Last edited by Random Variable; December 21st 2011 at 12:34 PM.
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