Sometimes one can calculate a definite integral such as (or more generally , where is a measurable set, is an integrable function, and is a measure) exactly by antidifferentiating and using the fundamental theorem of calculus. However, sometimes this method does not work, or is not appropriate. (The second might be the case if, for example, one was integrating not an explicit function but an unspecified function about which one had certain information.) Also, in many applications one does not actually need to compute an integral exactly; merely obtaining a sufficiently good approximation to that integral, or even just an upper bound on its magnitude, may be sufficient in many applications (particularly those in analysis).
This article gives links to further articles about techniques for approximating or bounding integrals. It is useful to make a distinction between non-oscillatory methods, which do not try to exploit cancellation in the sign or phase of the integrand , and oscillatory methods, which are designed to take advantage of such cancellation. For unsigned integrals, one should of course try the non-oscillatory methods; for signed integrals, the non-oscillatory methods tend to be easier but cruder (giving worse bounds), while the oscillatory methods give better bounds but are harder to implement.