Let $T(t)$ be a semigroup of bounded linear operator on a Banach space $X$. When does the following hold $$ \int_0^t T(s)x ds = (\int_0^t T(s) ds)x, x \in X \, , $$ where $ t \in (0,1)$.

Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold $$ \int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, , $$ where $ t \in (0,1)$?

Let $T(t)$ be a semigroup of bounded linear operator on a Banach space $X$. When does the following holds $$ \int_0^t T(s)x ds = (\int_0^t T(s) ds)x, x \in X \, , $$ where $ t \in (0,1)$.

Let $T(t)$ be a semigroup of bounded linear operator on a Banach space $X$. When does the following hold $$ \int_0^t T(s)x ds = (\int_0^t T(s) ds)x, x \in X \, , $$ where $ t \in (0,1)$.