Expansion of $ e^{\int f(t)\,dt} $

Question

What is the correct expansion of $ e^{\int f(t)\,dt} $ ?

Is it the following?

$$ \begin{split} e^{\int f(t)\,dt} & \simeq 1+\int f(t)\,dt + \frac{1}{2}\left( \int f(t)\,dt\right)^2\\ & =1+\int f(t)\,dt + \frac{1}{2}\iint f(t) f(t')\,dt\,dt' \end{split} $$

If yes, what are the conditions in the integral that this is true?

Answer 1

In general, the expansion of $e^x$ converges on all $\mathbb{C}$. The error of truncating the series is given by Taylor's Theorem. With regards to your integral it looks ok assuming your $f$ is not some funny function like a Gaussian or $x^x$ and has a manageable number of discontinuities.