This does Bayesian integration of functions of the form:
Where $d$ is the dimensionality of the space (so $x$ is $d$ dimensional), $f(x)$ is the function of interest and $g(x)$ is a pdf representing the density of each $x$ value.
This package uses the term Bayesian Integration to mean approximating a function with a kriging metamodel (aka a gaussian process model) and then integrating under it. A kriging metamodel has the nice feature that uncertainty about the nature of the function is explicitly modelled (unlike for instance a approximation with Chebyshev polynomials) and the Bayesian Integral uses this feature to give a Gaussian distribution representing the probabilities of various integral values. The output of the bayesian_integral_gaussian_exponential function is the expectation and variance of this distribution.