hello please there is someone who has an idea

Is that can help me

For any problem, where g[x] is independent of y and does not contain y and g[x] is "nice" and does not go to infinity, etc., is it true that

Integrate[g[x],{y,a,b}]==g[x]*(b-a)

If that is the case then can

Integrate[Integrate[f[s]^(1/n), {s, 0, y}], {t, 0, y1}] + 
 Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y1, y2}] + 
 Integrate[Integrate[(-f[s])^(1/n), {s, y, y1}], {t, y1, y2}] + 
 Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y2, 1}] + 
 Integrate[Integrate[(-f[s])^(1/n), {s, y2, y1}], {t, y2, 1}] + 
 Integrate[Integrate[f[s]^(1/n), {s, y2, y}], {t, y2, 1}]

be simplified to

Integrate[f[s]^(1/n), {s, 0, y}] y1 -
 Integrate[f[s]^(1/n), {s, 0, y1}] (y1 - y2) -
 Integrate[(-f[s])^(1/n), {s, y, y1}] (y1 - y2) +
 Integrate[f[s]^(1/n), {s, 0, y1}] (1 - y2) +
 Integrate[(-f[s])^(1/n), {s, y2, y1}] (1 - y2) +
 Integrate[f[s]^(1/n), {s, y2, y}] (1 - y2)

If this is correct then this might be a somewhat easier problem.

Integrating a fractional exponent of a quadratic equation seems to be a difficult problem for Mathematica.

Is anything known about the value of n or is it completely unknown? If more information were available about that then this might help.

Is there anything else that is known about the problem that might make it easier to solve?

From his Word to Mathematica:

I suppose that y1 <= y2

A function U is written

u[s_] := Piecewise[
  {{Integrate[f[s]^(1/n), {s,0,y}], 0<=s<=y1},
   {Integrate[f[s]^(1/n), {s,0,y1}]+Integrate[-f[s]^(1/n), {s,y,y1}], y1<=s<=y2},
   {Integrate[f[s]^(1/n), {s,0,y1}]+Integrate[-f[s]^(1/n), {s,y2,y1}]+Integrate[f[s]^(1/n), {s,y2,y}], y2<=s<=1}}]

such as

f[s_] := 1/2 (s^2 - (y1 + y2) s + y1*y2);

I think if I apply this condition

Integrate[u[s], {s, 0, 1}] == 0

I find

Integrate[Integrate[f[s]^(1/n), {s, 0, y}], {t, 0, y1}] +
  Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y1, y2}] +
  Integrate[Integrate[(-f[s])^(1/n), {s, y, y1}], {t, y1, y2}] +
  Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y2, 1}] +
  Integrate[Integrate[(-f[s])^(1/n), {s, y2, y1}], {t, y2, 1}] +
  Integrate[Integrate[f[s]^(1/n), {s, y2, y}], {t, y2, 1}] == 0

I try to solve this equation to find a relationship between y1 and y2 with y1 and y2 are the roots of f[s]

If you write a very simple clear explanation of the problem that you are having, something that anyone who knows nothing about all the time you have already put into your problem will still be able to understand exactly what you have and what you need to get, and you attach that file to your post (using the "add file to this post" button that is at the bottom of the Reply to this discussion window) then any reader here might take a few minutes to look at your file and offer some assistance.

I unable to write the equation in this producer, I send it as a Word file by e courier to Moderation Team