Hi! This integral is apparently easy but very difficult. My professor gives a solution (with explanation) which is totally different from solutions that you can find on wolfram alpha, symbolab etc.
Enjoy!

It does have negative area. The function is negative for 0< x < 1 , 0 for x=1 and positive for x>1. The negative area is just larger than the positive area.

I found an answer, barring a little bit of hand waving once I got to the functional equation.does anybody know how to rigorously show that my solution is the only solution.

How can this graph have a negative area?

it hasn't have that ahah

It does have negative area. The function is negative for 0< x < 1 , 0 for x=1 and positive for x>1. The negative area is just larger than the positive area.

you're right sorry!!! I don't know why but in my mind it was from infinite to 0

@Luca Sandrin Not to worry. This kind of thing has happened to me several times. We've all done it.

Looks like a tough one. Maybe Leibniz Method?

@Ian Fowler After one year I still can't find a working method for that integral ahahahah

I found an answer, barring a little bit of hand waving once I got to the functional equation.does anybody know how to rigorously show that my solution is the only solution.