Existance of the integral in the domain of generator of the strongly continuous semigroup

Question

Let $\{s(t)\}_{t\geq 0}$ is a $C_0$ semigroup of bounded operator on the Banach space $X$ and $A:D(A)\subset X\rightarrow X$ be the infinitesimal generators of the semigroup $\{s(t)\}_{t\geq 0}$. Therefore the operator $A$ is densly defined and closed operator. Let $g\in L^1(0,T;D(A))$,then is it true that $\int_0^T s(T-p)g(p)dp$ will be in $D(A)$? ${\int_0^T||s(T-p)g(p)||dp}$ is finite that part is ok. Help me