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Numerically integrates an expression over a range.
`NIntegrate` on well-behaved finite ranges agrees with the reference. The known failure class is oscillatory integrands over infinite ranges, where it can return a wildly wrong value while reporting success. Two ways to check an answer you rely on: evaluate the same integral over two finite truncations of the range and compare - they should agree with each other and with the infinite-range value - or compute the integral symbolically with `Integrate` when a closed form exists. On the Fresnel case above the symbolic route is exact where the numeric one fails. A finite range is not automatically safe: an integrand that oscillates unboundedly towards an endpoint can make the whole-range answer disagree with the sum of its own halves, which is the cheapest self-check an integrator has and costs one extra evaluation. ```wolfram NIntegrate[(1/x) Cos[Log[x]/x], {x, 0, 1}] (* 2.5186 *) NIntegrate[(1/x) Cos[Log[x]/x], {x, 0, 1/2}] + NIntegrate[(1/x) Cos[Log[x]/x], {x, 1/2, 1}] (* 2.2398 - 11% apart *) ``` Neither call warns. When the two disagree, at least one is wrong and the disagreement does not say which - treat the value as unverified rather than picking the one you prefer.
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