I am given that for $A$ that is $n \times n$ matrix of full rank,
$\int_{0}^{t}e^{A\sigma}d\sigma = (e^{At}-I)A^{-1}$
Then I am using this to solve LTI system
$\dot{x}=Ax+Bu$
Here, $x(0) = x_{0}$ and u is a constant vector.
I went ahead and used the general solution for LTI system,
$x(t)=e^{A(t-t_{0})}x_{0}+\int_{t_0}^{t}e^{A(t-t_{0})}B(\tau)u(\tau) \, d\tau$
I have $B$ and $U$ constant and time from 0 to t so this reduces to
$x(t)=e^{At}x_{0}+\int_{0}^{t}e^{A(t-t_{0})}Bu \, d\tau$
I am kinda stuck here, what should I do with those constant matrix $B u$ to solve this system using $\int_{0}^{t}e^{A\sigma}d\sigma = (e^{At}-I)A^{-1}$ ?
I know I am not allowed to just pull out $Bu$ outside of the integral because I am dealing with matrices. Any ideas?