$A$ and $B$ are square matrices
$$\exp(t(A+B))=\exp(tA)+\int_0^t \exp((t-s)A)B\exp(s(A+B))\,\mathrm ds$$
I found it from problem sets. It seems to define a new kind of matrix operation. I can check its truth when $A,B$ are scalars ($n=1$), but fail to prove it for general cases. The matrices do not commute and I cannot think of a way to do it. Can anyone give me a solution?