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For small $n$ Monte Carlo integration is not needed. For $n$ up to 100 see Kolmogorov-Smirnov Tests when Parameters are Estimated with Applications to Tests of Exponentiality and Tests on Spacings (t …

An overview by one of the developers of Mathematica, focusing on definite integrals, is at Symbolic definite integration: methods and open issues. Mathematica knows all the entries in Gradshteyn-Ryzhi …

Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_ …

For large $s$ a single sum in terms of a hypergeometric function may be useful, $$c_{p}=\frac{(-1)^p}{2^p p!}\sum_{n=0}^\infty\frac{(n)_p}{(2s)^{n}n!}(m+1)^{n-p} {}_2F_1\bigl(p+1/2,p-n,2p+1,2/(m+1)\bi …

Q: The OP seeks a "reasonable approximation" for large or small $\rho_0$ of the function $$P(x)= \int_0^{x} e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0)\,d\rho,\;\;x\geq 0.$$ Consider the kernel $$p …

You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be A stochastic algorithm for high-dimensional integrals over unbounded regions with Gaussian weight …

You want the relation between $\phi(x)$ evaluated at a point $x$ on the boundary and the derivative $\partial\phi/\partial n$ evaulated at the same point $x$, which is given by $$\phi(x)=\frac{\partia …

Since a B-spline is a piecewise polynomial function, the question is whether there exists an exact equality formula for the integral $\int_{-a}^{b}u^pe^{-u^2/2}du$. This integral equals an elementary …

The limit is $$\lim_{n\rightarrow\infty} \int_0^{\pi/2} \frac{e^x}{\left(2x/\pi\right)^n + 1} \left| \sin(x) \sin(nx) \sin(2nx) \right| \, dx=\frac{2 \left(1+e^{\pi /2}\right)}{3 \pi }=1.23302\cdots.$ …