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Math Help - Prove the Integral

  1. #1
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    Prove the Integral

    Prove that the function: l1-2xl has a Integral in the section(see in the picture)

    thanks.
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  2. #2
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    Quote Originally Posted by ZOOZ View Post
    Prove that the function: l1-2xl has a Integral in the section(see in the picture)

    thanks.


    Did you mean to prove that the improper integral \int\limits^\infty_{-\infty}|1-2x|\,dx exist? Because this integral

    does not exist (i.e., it's divergent), since |1-2x|=1-2x\Longleftrightarrow x\leq\frac{1}{2} , so

    we can write \int\limits^\infty_{-\infty}|1-2x|\,dx=\int\limits^{1/2}_{-\infty}(1-2x)\,dx+\int\limits^\infty_{1/2}(2x-1)\,dx , and both these integrals diverge.

    Tonio
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  3. #3
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    hi i mean how i prove that this function have a Antiderivative (in the same section)

    thanks.
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  4. #4
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    Using what basis? That's a continuous function and all continuous functions are integrable.

    If you do not know that theorem, then the simplest way to show a function is "integrable" is to find its integral!

    If x< 1/2, |1- 2x|= 1- 2x. Integrate that, including the "constant of integration" C. If x> 1/2, |1- 2x|= -(1- 2x)= 2x- 1. Integrate that, including the "constant of integration" D. At x= 1/2, those must be the same, allowing you to write either C in terms of D or D in terms of C so that you have only a single "constant of integration".
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