Let's asumme some integrable function $f (t,x)$ with respect to $t $ and some parameter $x,a,b \in \mathbb{R} $. In such way $\int_b^a f (t,x)dt =F (a,x)-F (b,x) $
Let's assume that we know closed form solution for $F (x,a) $ and $F (x,b) $. Is it true that equation will holds true for all square n×n matrices $X, A,B$ written instead of $x,a,b $
example: $\int_b^a t^x dt =\frac {a^{x+1}-b^{x+1}}{x+1} $. You can write that solution as Taylor series so for matrix substitution it will have some vlaid results. For example $e^A =\sum_{k=0}^{\infty} \frac {A^k}{k!}$ is treat as definition. In such way what I want to ask in that scenario is it sufficient to get closed form solution of integral to write
$\displaystyle \int_b^a t^x dt =\frac {a^{x+1}-b^{x+1}}{x+1} \Rightarrow \int_B^A t^X dt=\frac {A^{X+1}-B^{X+1}}{X+1} $
PS: My knowledge about matrices is $\epsilon>0 $ so if there are any issues please correct me.
The answer is no, because if you want to take into account matrix-dependent functions you have to readapt the meaning of the boundary of your integration interval. To be clearer, in your case you have that $\partial[a,b]=\{a,b\}$. If you let $T$ belongs to $\Omega\subset\mathbb{R}^{n\times n}$ for example you have to expect to obtain something related to the boundary of $\Omega$. The generalisation you are looking for is the "Generalized Stokes theorem". And possible restrictions regard the regularity and the nature of $\partial\Omega$.