Inverse operator and boundness

[Солунац]

Problem: In space $C[0,1]$ is defined an operator A as
$A(x(t))=x(t)-\int\limit_0^1 tsx(s) ds , $ $ 0\leqslant t\leqslant 1$.
It is linear, bound and bjective. Find $A^-^1$ and prove that it is bound too.

From $Ax=y$ is $x(t)=y(t)+C\cdot t$ where is $C=\int_0^1 sx(s) ds$. But I am stuck here, so please help. I need to express $C$ in terms of $s$ and $y(s)$ to find $A^-^1$, but I don`t have an idea how to do it.

[V.V.]

Find $C$.
Multiply by $t$ and integrate by $[0,1]$.

$\int\limits_0^1 tx(t)\,dt=\int\limits_0^1 ty(t)\,dt+C\int\limits_0^1 t^2\,dt$;
$C=\int\limits_0^1 ty(t)\,dt+\frac{C}{3}$;
$C=\frac{3}{2}\int\limits_0^1 ty(t)\,dt$.

[Солунац]

Спасибо V.V.