I would like to thank in advance anyone willing to take a look at my question.
I am asking whether there exists a method which can be used to numerically evaluate an integral of a function containing several parameters. More specifically, we want to evaluate the integral of function $f = f(x,p_{1},p_{2},...,p_{n})$ over the variable $x$, where {$p_{1},p_{2},...,p_{n}$} are some parameters which appear in the function. Is there an known algorithm/numerical method which enables you to obtain a value of the integral in terms of these parameters?
For anyone more interested, I would specifically like to integrate a function of the form:
$$
F(x,p_{1},p_{2},...,p_{n})= \frac{e^{-\frac{(x-f(x,p_{1},p_{2},...,p_{n}))^{2}}{(g(x,p_{1},p_{2},...,p_{n}))^{2}}}}{\sqrt{\pi}g(x,p_{1},p_{2},...,p_{n})}\,
$$
which you could use to describe a normal distribution whose mean and variance are not constant, but rather functions of the random variable.
You could try something like Simpson's Rule for a fixed $n$, obtaining a rather complicated function of the parameters that (under suitable conditions) might be a good approximation to the actual value, at least in some region of parameter space.
Another possibility is to approximate the integrals numerically for some set of values of the parameters, and then try to fit this to a suitable function (say a low-degree polynomial) of the parameters.
A third possibility is to expand your integrand as a power series in the parameters, and numerically integrate the coefficients, obtaining some terms of a power series for the integral.