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Math Help - Definite Integral Involving Absolute Value

  1. #1
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    Definite Integral Involving Absolute Value

    I am having trouble understanding the following definite integral:
    \int_{-2}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - |x + 1|

    The way I did it is as follows:
    \int_{-2}^{0} 5 - x^2 + x + 1 + \int_{0}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - x - 1
    from which I got a similar answer to the correct one which looked as follows:
    \int_{-2}^{-1} 5 - x^2 + x + 1 + \int_{-1}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - x - 1

    My question is that due to it being an absolute value surely you have to break the integral up into \int _{a}^{0} + \int _{0}^{b}?
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  2. #2
    MHF Contributor

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    Why are you breaking the integral at x= 0? At x= 0, |x+ 1|= |1|= 1 and there is no problem. |a| changes "formula" when a= 0 so |x+ 1| changes when x+ 1= 0 or x= -1. Break the integral into \int_{-1}^{-1}+ \int_{-1}^{\frac{-1+\sqrt{17}}{2}.
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  3. #3
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    Oh damn, I was treating |x + 1| as |x| thanks for the help!
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