Let $T(t)$ be a $C_0$-semigroup on a Hilbert space $H$ with a generator $A$.
It is well known that for all $x\in H,$ we have: $ \int_0^t T(s)x ds \in D(A) $ and $ A\int_0^t T(s)x ds = T(t)x-x$.
How is this formula changed when $x$ is replaced by a continuous function $t \mapsto x(t)$ such that: $x(t)\in D(A),\, \forall t\ge 0?$