Let $A$ be a square matrix. Invertibility of $\exp(A)$ follows easily from properties of the matrix exponential.
Is $\int_0^t \exp(A u)du$ also invertible? I believe it should be, and that the inverse should be $I - At/2 + A^2t^2/12 + ...$
This comes from expanding the real-valued function $x/(e^x - 1)$ in a power series about $x=0$. How should I approach a proof of this (or could I find it in a book somewhere?)
What about the more general case when $\Phi(t)$ is defined by
$\frac{dX}{dt} = A(t)X,\ \ X_0 = x_0 $
and
$X(t) = \Phi(t)x_0$? How might one show that $\int_0^t \Phi(u)du$ is invertible?