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]

Here are the contents of his word document roughly translated into Mathematica for those who don't want to spend the time to try to open and read his file:

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

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

Such as

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

Now to try to address his problem I first start with a simpler problem and use Mathematica

f[s_] := 1/2 (s^2 - (y1 + y2) s + y1*y2);
Integrate[f[s]^(1/n), {s, 0, y}]

That takes long enough that I do not wait for it to finish. I presume the issue is his n'th roots of his quadratic equation in s. Without further explanation and clarification I don't see a way to approach this directly.

Since he refers to y1 and y2 being roots of his quadratic, I look at those roots. If I assume y1 and y2 real then the roots of his quadratic are either y1 or y2, depending on which of the two solutions are chosen and whether y1 is less than or equal to y2 or not.

I am puzzled by his outer integrals being over t and not over s. Those outer integrals then seem to only multiply the inner integrals by the difference in the limits of integration over t. Is that what he really meant this to be? If so then that seems to simplify his problem slightly.

From the tags he selected in his original post I presume he is trying to accomplish all this within WolframAlpha. That might seem to only compound the difficulty in finding the solution he is looking for.

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