So, I have the following integral : $$I = \int_0^t e^{-a(t-t')}f(t') dt'$$
If $a > 0$ is large enough, I know you can approximate the exponential by a dirac function, which gives $e^{-a(t-t')} \sim \frac{\delta(t-t')}{a}$, and the integral becomes $I = \frac{f(t)}{a}$.
However, this development is quite strong, and does not apply if $a$ is not that large. The question is, is there a better development at higher order for such integrals ?
I should add that we know almost nothing of $f$, except that it's continuous, and is at least $C^1$.
One option you certainly have, that is frequently employed by physicists, is to write the exponential $\exp(-a(t^\prime - t))$ as a Taylor-series and cut off lower order terms. Say you would just keep the first two or so. In many cases that will give you a reasonable approximation of the integral.
Alternatively, since $f$ is continuous you could have as a very rough estimate: $$ \int_0^t \exp(-a(t-s))f(s)\mathrm{d}s \le \frac{1 - exp(-at)}{a}\|f\|_\infty$$
where $\|f\|_\infty = \sup_{s \in [0,t]} f(s)$.