Find the power of operator I have an operator $U=x(t) + \int_0^1 x(st) \, ds$.
Goal is to find $U^2$ without iterated integrals.
I start with:
$$U(Ux(t))=x(t) + \int_0^1 x(st)\,ds+\int_0^1 (x(qt) + \int_0^1 x(sqt)\,ds )\, dq= \\ =x(t) + 2\int_0^1 x(st) \, ds + \int_0^1 \int_0^1 x(sqt) \, ds\,dq.$$
I have a problem in transformation $\int_0^1 \int_0^1 x(sqt) \, ds\,dq$ . I have used integration by parts for the internal integral, but i do not know what to do next.

Let $s=\frac{u}{q}$, $ds=\frac{1}{q}\,du$, and integrate by parts
\begin{align}
    \int_0^1 \int_0^1 x(sqt)\,ds\,dq &= \int_0^1 \frac 1 q \int_0^q x(ut)\,du\,dq \\
     & = \left.\ln q\int_0^q x(ut)\,du\right|_{q=0}^{q=1}-\int_0^1 (\ln q)x(qt) \, dq \\
   & = -\int_0^1 x(qt)\ln q \,dq.
\end{align}
