The images below are about Darboux Integral Definition (written as a theorem in the last picture, as our initial definition was the Riemann definition)
The Darboux Theorem statement, I refer to is as follows:
Now, in the definition, it is said that the supremum and infimum are taken over the set of $L(f,P)$ and $U(f,P)$ for all partitions. But in the first and second image, it is written that we take only a particular type of partitions, which divide the interval equally. But it is obviously not necessary that the supremum for the set of such specific partitions is the supremum for the set of all partitions. Then how are we using the theorem, to compute the integral of $x$?
It turns out that it is sufficient to consider only these partitions when the limits of the upper and lower sums over these partitions exist and are equal.
Note that
$$ L(f,\mathcal P_n) \le \sup_{\mathcal P} L(f,\mathcal P) \le \inf_{\mathcal P} U(f, \mathcal P) \le U(f,\mathcal P_n) $$
for any partition $\mathcal P_n$. Hence, if we have that
$$ \lim_{n\to \infty} L(f,\mathcal P_n) = \lim_{n \to \infty} U(f,\mathcal P_n), $$
then it must also be the case that
$$ \sup_{\mathcal P} L(f,\mathcal P) = \inf_{\mathcal P} U(f, \mathcal P). $$