Most integrals that arise in application cannot be evaluated analytically. It’s satisfying when integrals can actually be evaluated in closed form, though in general the real world is not as easy on us as calculus homework assignments lead students to expect.
Sometimes it’s possible to evaluate an integral in closed form by expanding your idea of what closed form means, going beyond the most familiar functions and using special functions. The advantage of reducing your integration problem to special functions is that these functions have been studied extensively, and high-quality software exists for numerically computing them.
Difficult integrals may yield to common numerical integration techniques, though usually some preparation is needed in order to transform the integral into the form that the technique handles best.
When standard techniques do not work—maybe they are not accurate enough or fast enough—it may be necessary to create new integration algorithms. (The line between creating a new algorithm and customizing standard algorithms is blurry.) No algorithm can reliably integrate everything. Often it’s possible to construct custom methods that take advantage of the specifics of a problem.
It can be risky to use off-the-shelf integration software without some understanding of both the integration problem and the algorithm; the software may confidently return a wrong result. But with some understanding of the problem at hand, it may be possible to determine upper or lower bounds on the correct answer and thus have a way to test the plausibility of the computed answer. Depending on the context, these upper or lower bounds may be good enough to eliminate the need to actually compute the integral.
Statistics, especially Bayesian statistics, is a rich source of challenging integration problems. Sometimes it is not possible (or necessary) to compute the integral per se. It may be enough to draw samples from a probability distribution associated with the integral using Markov Chain Monte Carlo (MCMC). While MCMC makes it possible to approach problems that would be intractable otherwise, MCMC methods present their own challenges.
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