The accepted answer to this question, makes the claim that for matrices $A$ and $X$, where $X$ is positive definite Hermitian,
$$ \int_0^{\infty}\frac{1}{1+tX}A\mathrm{d}t\frac{1}{1+tX}=\int_0^{\infty}\mathrm{d}t\int_0^1\mathrm{d}s \mathrm{e}^{-stX}A\mathrm{e}^{stX}\mathrm{e}^{-tX}\text{.} $$
However, I don't think this is true. I worked out the Taylor expansion of both expressions and carried out the integral over $s$, the results seem to differ by multiplicative factor in each terms of the series sum.