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Homework Help: Improper Integral

  1. Mar 20, 2006 #1

    G01

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    [tex] \int_{-\infty}^{\infty} e^{-|x|} dx [/tex]

    Could someone tell me why this integral, when you split it comes out to be:

    [tex] \int_{-\infty}^0 e^x dx + \int_0^{\infty} e^{-x} dx [/tex]

    I keep thinking it should be e^(-x) in the first integral. I don't know why its positive. I can solve this integral otherwise. Thanks again.
     
  2. jcsd
  3. Mar 20, 2006 #2
    The absolute value is defined for a real number [itex] x [/itex] by
    [tex]
    |x|= \left \lbrace
    \begin{array}{l l}
    x & \mbox{if} \ x \geq 0 \\
    -x & \mbox{if} \ x\leq 0
    \end{array}
    \right.
    [/tex]

    when [itex]x[/itex] is in the interval [itex](-\infty,0)[/itex], then [itex] |x| = -x [/itex]. Likewise, when [itex]x[/itex] is in the interval [itex](0,\infty)[/itex], then [itex] |x| = x [/itex]. Therefore,

    [tex] \int_{-\infty}^0 e^{-|x|}dx = \int_{-\infty}^0 e^{-(-x)}dx
    = \int_{-\infty}^0 e^x dx [/tex]
     
    Last edited: Mar 20, 2006
  4. Mar 20, 2006 #3

    G01

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    Im sorry that definition confuses me...... I always thought the absolute value was the positive distance of that number from the origin, leading me to believe that |-x|= x. This seems contrary.
     
  5. Mar 20, 2006 #4
    This is true.
    This is false. For example take [itex]x=-3[/itex]. Then your saying that [itex]|-(-3)| = -3[/itex], which is the same as saying that [itex]|3| = -3[/itex], which is clearly false. I used paranthenses to delimit the value which was substituted for x, namely -3.

    However, if we use the definition of absolute value, we arive at the correct result. For example, if [itex] x=-3[/itex], we have that [itex]x<0[/itex] so we use the second case of the definition to arrive that [itex]|x| = -x[/itex], which gives [itex] |-3| = -(-3) = 3[/itex].

    Try plugging in different values for x into the definition to convince yourself that this definition works.

    The definition for when [itex]x>0[/itex] should be clear. The absolute value of a positive number is a positive quantity and is equal to that number. However if [itex] x<0 [/itex], then [itex] -x>0 [/itex], so [itex]-x[/itex] is a positive number (don't let the negative sign confuse you, remember that the negative of a negative number is positive).
     
    Last edited: Mar 20, 2006
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