Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion:
Let $A(t)$ be an $n\times n$ matrix of real valued functions. Then $\int_0^t \left(A(x)A(t)-A(t)A(x)\right) dx = \left(\int_0^t A(x)dx\right)A(t)-A(t)\left(\int_0^t A(x)dx\right)$