What is the best way to approximate a function of the following form,
$$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$
Any approximation to this, does taylor series work? The reason I am doing so is because $f(x)$ cannot be integrated in closed form.
If the form of $f(x)$ is needed then I will edit my question and write it down.
Appreciate any advice.
Assuming the improper integral exists, your function can be written as
$$F(y) = C \exp\left(\int_0^y f(x)\; dx\right)$$ where $C = \exp(-\int_0^\infty f(x)\; dx)$.
Assuming the required derivatives exist, the Taylor series of $F(y)$ around $y=0$ starts
$$ F(y) = C + C f(0)\; y + \dfrac{C}{2} \left( f'(0) + f(0)^2\right) y^2 + \dfrac{C}{6} \left( f''(0) + 3 f(0) f'(0) + f(0)^3\right) y^3 + \ldots $$
Is that what you're looking for?