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what is the integration of e^-|x| from negative infinity to x ?

  • MATH -

    since |x| = -x for x < 0.
    ∫[-∞,x] e^-|t| dt
    = ∫[-∞,x] e^t dt if x < 0
    = e^t

    So, for x>=0,
    ∫[-∞,x] e^-|t| dt
    = ∫[-∞,0] e^t dt + ∫[0,x] e^-t dt
    = 1 + (1-e^-x)
    = 2 - e^-x

  • MATH -

    What if we have .. integration of xe^(|x|) dx from negative infinity to x.

  • MATH -

    No idea. Do it the way I did, but you have to use integration by parts. If you get stuck, show how far you got.

    You should wind up with

    -(x+1)e^-x for x<0
    (x-1)e^x for x>=0

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