Is the function integrable by Lebesgue on a ray

RandomNumGenerator · Jan 13, 2023

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

Is the function integrable by Lebesgue on a ray

RandomNumGenerator

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