Is the function integrable by Lebesgue on a ray

Discussion in 'Differentiation and Integration' started by RandomNumGenerator, Jan 13, 2023.

  1. RandomNumGenerator

    RandomNumGenerator

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    Hello! I was very tormented with this Lebesgue-related problem. I searched the whole Internet, but I didn't find a solution (maybe I was looking in the wrong places). The task looks like this:

    Is the function integrable by Lebesgue on a ray (-∞,-1)?
    upload_2023-1-13_10-3-13.png
    if it is, then find its value.


    I've tried to find Riemann improper integral
    upload_2023-1-13_10-8-56.png
    upload_2023-1-13_10-9-5.png
    upload_2023-1-13_10-9-14.png
    upload_2023-1-13_10-9-20.png
    upload_2023-1-13_10-9-27.png
    upload_2023-1-13_10-9-50.png
    But in the end, there happens division by zero (in first addendum)
    upload_2023-1-13_10-10-56.png


    I'm not asking for a complete solution, but at least a hint in which direction to move, which theorems to use, otherwise I'm completely desperate. Thanks!



    PS I'm sorry if there is some wrong terminology - I'm not native english speaker
     
    Last edited: Jan 13, 2023
    RandomNumGenerator, Jan 13, 2023
    #1
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