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Definite integral of an absolute value function

  1. Jun 25, 2015 #1
    Can we integrate:
    $$\int_a^b |x| dx$$
    using an antiderivative of ##|x|##, namely ##\frac{1}{2} x |x|##, instead of splitting up the integration interval?
    I know this is not particularly useful for integrals such as:
    $$\int_{-5}^5 |t^3 - 8| dt$$
    However, for absolute value functions with linear arguments, this method (if valid) would be much more efficient.
     
  2. jcsd
  3. Jun 25, 2015 #2

    HallsofIvy

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    Yes, of course. If F(x) is an anti-derivative of f(x) then [itex]\int_a^b f(x) dx= F(b)- F(a)[/itex]. That is true for f(x)= |x| and F(x)= (1/2)x|x|.
     
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