Trace of Integral

Trace of Integral

Given a n by n matrix $A$, where $a_{ii} = 1$ $\forall i$, I was wondering
if the following calculation is possible with trace, \begin{equation} tr
\int_0^1 e^{tA} dt = \int_0^1 tr(e^{tA}) dt \end{equation} We know that
$tr(e^{tA})= e^{tr(tA)}$ $\Rightarrow$ $$\int_0^1 tr(e^{tA}) dt= \int_0^1
e^{tn} dt= \frac{e^n - 1}{n} $$ I'm not confidant about the first equality
(bringing trace inside integral)