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Thread: Integration

  1. #1
    ixi
    ixi is offline
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    Integration

    The question is:

    The integration of e^(-abs (x)) from -infinitiy to + infinity



    The absolute before the x is throwing me off...

    I got

    -e^(-abs(x)) from -infinitiy to + infinity

    i graphed it but doesn't it go to 0?

    so would the answer be 0?


    thanks
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  2. #2
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    $\displaystyle \int_{-\infty}^\infty e^{-|x|}dx=\int_{-\infty}^0 e^{-|x|}dx+\int_{0}^\infty e^{-|x|}dx$

    Now when $\displaystyle x\in(-\infty,0], |x|= -x,$ so $\displaystyle -|x|= -(-x)=x$

    And when $\displaystyle x\in[0,\infty), |x|=x$ and so $\displaystyle -|x|=-x$

    So therefore $\displaystyle \int_{-\infty}^0 e^{-|x|}dx+\int_{0}^\infty e^{-|x|}dx=\int_{-\infty}^0 e^{x}dx+\int_{0}^\infty e^{-x}dx$

    $\displaystyle =1+1=2$ if you need help with the integration let me know
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