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!

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.

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.

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.

How can this graph have a negative area?